The School Bell SG — Timely updates for students and parents

The CNA article highlights the growing concerns over aggressive and sometimes unethical marketing tactics used by tuition centers in Singapore, particularly in subjects like secondary school mathematics, where demand is high. If you’re a parent considering tuition for your child, here are the key takeaways:

1. The Tuition Boom and Parental Anxiety

The tuition industry has seen a sharp increase in household spending, rising from S$1.1 billion in 2013 to S$1.8 billion in 2023. Many parents feel pressured to enroll their children in extra classes to keep up with academic competition. For subjects like secondary school math, where structured problem-solving is crucial, tuition is often seen as necessary for ensuring good grades.

2. Selective Enrollment and “Guaranteed A” Promises

Some tuition centers market themselves as “elite” by selectively enrolling only students who already perform well in math. This allows them to claim high success rates and promote “guaranteed A” results. However, this means that students who struggle the most—those who might genuinely need tuition—could be rejected.

3. Fear-Based Marketing Tactics

Tuition centers often exploit parental fears by emphasizing how not enrolling in tuition could lead to poor grades and limited future opportunities. Advertisements highlight high-stakes exams like the O-Levels, subtly suggesting that students who don’t get tuition will fall behind. Some centers even use comparisons of “before and after” resultsor highlight exaggerated “miracle transformations” in student performance.

4. Referral Incentives and High Costs

Many tuition centers rely on word-of-mouth referrals by offering discounts or cash rewards to parents who bring in new students. This creates a competitive atmosphere where parents feel compelled to recommend tuition services, even if they’re not entirely convinced of the effectiveness. Meanwhile, the cost of tuition continues to rise, with premium centers charging hundreds of dollars per month.

5. Government Response and Call for Ethical Practices

Education Minister Chan Chun Sing has acknowledged these issues, stating that the Ministry of Education (MOE) is working with the advertising industry to develop a code of conduct for tuition center marketing. However, the root problem—Singapore’s academic pressure and parents’ perception that tuition is essential—remains a challenge.

What This Means for Parents Looking for Secondary Math Tuition

• Be skeptical of “guaranteed A” claims – No tuition program can ensure success without effort from the student.
• Watch out for selective enrollment tactics – Some centers might reject weaker students to maintain their success rates.
• Avoid fear-based pressure – Your child may need tuition, but choosing the right tutor matters more than rushing into a class out of anxiety.
• Compare costs and value – Some expensive tuition programs are priced high due to branding rather than actual teaching effectiveness.
• Look for reviews from real students and parents rather than marketing materials.

Ultimately, while tuition can be helpful for math improvement, parents should focus on quality teaching over flashy marketing claims and ensure that their child’s individual learning needs are met.

Direct School Admission System in Singapore’s Education System

The Direct School Admission (DSA) system is a pivotal component of Singapore’s education framework, offering students an alternative pathway to secure placements in secondary schools and junior colleges (JCs) based on their talents in sports, arts, leadership, and academic niches. Introduced in 2004 by the Ministry of Education (MOE), the DSA aims to reduce overreliance on academic grades by recognizing holistic development¹³. Over two decades, the scheme has evolved significantly, though it has also faced critiques regarding equity and accessibility. As of 2025, the system remains under review to address emerging challenges while preserving its original mission of fostering diverse student potential²⁵.

Historical Development 

Origins and Implementation 

The DSA was first launched in September 2004 as DSA-Secondary (DSA-Sec), enabling Primary 6 students to gain early admission to secondary schools before taking the Primary School Leaving Examination (PSLE)¹³. This initiative emerged alongside broader reforms to Singapore’s education system, which sought to value non-academic talents and reduce stress associated with high-stakes exams. In 2005, the scheme expanded to include DSA-Junior College (DSA-JC), allowing Secondary 4 and 5 students to secure JC placements prior to their O-Level examinations³.

A subcategory, the School-Based Excellence Initiative (often termed the “Niche Scheme”), further enabled schools to recruit students excelling in specific non-academic domains, such as robotics or debate³. These changes reflected MOE’s commitment to creating a “flexible and broad-based” system that celebrated diverse forms of achievement¹.

Policy Revisions and Public Discourse 

By the 2010s, the DSA’s popularity grew, with approximately 20% of each student cohort utilizing the scheme by 2024². However, this expansion also sparked debates about fairness. Critics argued that wealthier families could disproportionately access resources like private coaching, thereby advantaging their children in DSA competitions⁵. In 2020, MOE introduced outreach programs to guide disadvantaged students through the DSA process, acknowledging that “those from less advantaged backgrounds may not be as well-informed”⁵.

Structure and Application Process 

DSA-Sec and DSA-JC Pathways 

The DSA-Sec allows Primary 6 students to apply to secondary schools via an online portal, submitting portfolios highlighting achievements in over 20 talent areas, from volleyball to robotics⁴⁶. Schools conduct trials, auditions, or interviews to assess applicants, with successful candidates receiving offers by September⁶. Similarly, DSA-JC enables O-Level students to secure JC spots through demonstrations of leadership or niche abilities³.

Once admitted, students must commit to their chosen school for their entire secondary or JC education and participate in activities related to their declared talent⁴. MOE emphasizes that the DSA is “not a pathway for every student,” cautioning that mismatches between a student’s academic readiness and a school’s rigor could lead to struggles⁵.

Key Dates and Logistics 

Applications for DSA-Sec typically open in May, with deadlines in early June. For the 2024 cycle, submissions closed on June 3, followed by school-specific selection processes⁶. Parents are advised to align their child’s strengths with a school’s specialized programs, as misalignment could jeopardize admission chances⁴.

Criticisms and Challenges 

Socioeconomic Disparities 

A persistent critique of the DSA is its perceived bias toward affluent families. Members of Parliament, including Denise Phua, have noted that children from higher-income households often benefit from private coaching, elite sports camps, and portfolio-building services—resources less accessible to lower-income groups⁵. This dynamic risks transforming the DSA into a system where “talent” becomes synonymous with “financial privilege,” undermining MOE’s equity goals²⁵.

“Green Harvesting” and Integrity Concerns 

In 2024, a bribery scandal involving a basketball coach highlighted vulnerabilities in the DSA framework. The coach allegedly charged up to S$50,000 to secure placements at prestigious schools like Anglo-Chinese Junior College, leading to his deregistration and criminal investigation². Such incidents fueled public skepticism about the transparency of selection processes, with Workers’ Party MP Jamus Lim calling for stricter oversight to prevent “gaming” of the system².

Additionally, the practice of green harvesting—where schools recruit students based on early potential rather than proven merit—has raised concerns about premature specialization and undue pressure on young children².

Recent Reforms and Future Directions 

2025 MOE Review 

In January 2025, MOE announced a comprehensive review of the DSA system, responding to critiques from policymakers and parents. Proposed reforms include:

1. Enhanced Transparency: Standardizing selection criteria across schools to reduce subjectivity.
2. Anti-Bias Measures: Implementing anonymous assessments during trials or auditions.
3. Safeguards Against Green Harvesting: Restricting schools from recruiting students below a certain age for specific talents2.

Education Minister Chan Chun Sing emphasised that the review aims to “refocus the DSA on student development rather than performance,” ensuring the scheme remains aligned with its original mission².

The Workers’ Party’s 10-Year Through-Train Proposal 

Associate Professor Jamus Lim has advocated for a 10-year through-train program, allowing students to progress from primary to secondary school without high-stakes exams at age 12. This model, already piloted in select schools, would grant students more time to explore interests before specializing, potentially reducing DSA-related pressures².Critics, however, argue that delayed streaming could hinder academic preparedness for national exams.

Impact and Legacy 

Over 300,000 students have utilized the DSA since its inception, with many attributing their success to the recognition of non-academic strengths³⁵. Notable alumni include national athletes and artists who credit the scheme for providing opportunities to excel. Nevertheless, the system’s legacy remains intertwined with ongoing debates about meritocracy, equity, and the purpose of education in Singapore.

Conclusion

The DSA system represents both the aspirations and complexities of Singapore’s education reforms. While it has diversified pathways for talent development, its future hinges on addressing structural inequities and adapting to societal needs. As MOE’s 2025 review unfolds, stakeholders await reforms that balance merit-based recognition with inclusivity, ensuring every student—regardless of background—can thrive.

Citations:

1. https://www.nlb.gov.sg/main/article-detail?cmsuuid=bf01ac01-4d4d-42e6-94d6-b117a4ed5f64
2. https://www.theonlinecitizen.com/2025/01/03/jamus-lim-welcomes-moes-dsa-review-urges-more-flexibility-in-education-system/
3. https://en.wikipedia.org/wiki/Direct_School_Admission
4. https://www.computhink.com.sg/latest-dsa-updates-for-concerned-parents/
5. https://www.ntu.edu.sg/nie/news-events/news/detail/commentary-is-dsa-becoming-almost-as-important-as-the-psle-in-the-transition-from-primary-to-secondary-school
6. https://www.moe.gov.sg/secondary/dsa
7. https://gutzy.asia/2024/08/02/wp-mp-louis-chua-urges-review-of-direct-school-admissions-scheme/
8. https://www.channelnewsasia.com/commentary/singapore-direct-school-admission-dsa-psle-secondary-school-students-sports-music-4480716
9. https://www.straitstimes.com/singapore/more-avenues-in-the-education-system-for-students-to-find-their-best-fit-chan-chun-sing
10. https://www.reddit.com/r/SGExams/comments/17qjv99/why_i_think_dsa_should_not_be_a_thing/
11. https://ahmadibrahimsec.moe.edu.sg/about-us/direct-school-admission/
12. https://www.straitstimes.com/singapore/mindsets-must-shift-for-changes-in-education-to-have-lasting-impact-chan-chun-sing
13. https://www.channelnewsasia.com/singapore/direct-school-admission-moe-investigate-any-allegations-unfair-practices-dsa-4105961
14. https://deyisec.moe.edu.sg/useful-links/direct-school-admission-dsa/
15. https://www.todayonline.com/singapore/moe-will-investigate-any-allegations-unfair-practices-direct-school-admission-exercise-chan-chun-sing-2358246
16. https://slcedu.sg/direct-school-admission-singapore/
17. https://www.channelnewsasia.com/watch/chan-chun-sing-moe-review-dsa-scheme-4678981
18. https://www.moe.gov.sg/secondary/dsa/eligibility
19. https://www.facebook.com/TheStraitsTimes/posts/more-details-of-the-changes-to-the-dsa-scheme-will-be-released-in-the-second-qua/899746155645879/
20. https://www.linkedin.com/posts/mrhugobear_review-of-dsa-scheme-among-moes-priorities-activity-7280749713786597376-fqQ6
21. https://www.instagram.com/dlsaa_sg/p/DETyajnh81C/
22. https://forum.kiasuparents.com/topic/105576/dsa-2025
23. https://www.linkedin.com/posts/alan-go-42690024_review-of-dsa-scheme-among-moes-priorities-activity-7280395502980886528-9617

Summary of Interview with Singapore’s Education Minister Chan Chun Sing

1. Education Philosophy & School Perceptions
   Minister Chan emphasizes moving beyond labels like “elite” vs. “neighborhood” schools. He stresses self-confidence and unique skill development, drawing parallels to the U.S. Special Forces’ diverse talent approach. Schools should help students discover their strengths, preparing them for real-world challenges like job interviews.
2. Lifelong Learning & SkillsFuture
   The focus shifts from rote learning to fostering joy in learning and adaptability. SkillsFuture credits aim to support adult learners, but Chan highlights time management as the bigger hurdle. He advocates for tech-enabled, bite-sized learning (e.g., “TeachTok”) to make education accessible and personalized.
3. Role of Teachers & Collaboration
   Teachers are not just knowledge transmitters but mentors who inspire curiosity. Chan underscores partnerships with industries and alumni to keep education relevant. He defends teachers’ professionalism, urging parents to respect their expertise despite higher education levels today.
4. Political Culture & Leadership
   Chan warns against divisive politics, advocating for unity of purpose over populism. He stresses the need for leaders with integrity and conviction, capable of tough decisions for long-term national good, akin to Singapore’s pioneers (e.g., NS, HDB policies).
5. Youth & Social Media Challenges
   Younger generations face pressure from social media comparisons, risking mental health. Chan encourages defining self-worth beyond likes/followers and fostering resilience. He critiques viral content overshadowing critical national issues (e.g., geopolitical risks).
6. Personal Reflections & Values
   • Handling Criticism: Chan shares struggles with online mockery but stays grounded through family support and visiting schools for inspiration.
   • Work-Life Balance: Prioritizes family time and self-reflection to transition from ministerial duties to being a father/husband.
   • Motivation: Driven by ensuring Singapore’s survival and inclusivity, especially for vulnerable groups like special needs individuals.
7. Vision for Singapore
   • SG100 Goal: A nation defying historical odds through global connectivity, innovation, and social cohesion.
   • Education’s Future: Lifelong learning, equity (e.g., every student’s potential), and fostering a “can-do” spirit.
   • Public Service: Seeks leaders with honesty, courage, and a heart for service, ensuring Singapore remains a meritocratic, compassionate society.

Key Quotes & Metaphors

• On Resilience: “If you don’t look down on yourself, nobody will look down on you.”
• On Politics: “Success is enabling the next generation to surpass us, not personal accolades.”
• On Everton FC: Used to illustrate valuing underdogs and nurturing potential despite limited resources.

Children’s arithmetic skills do not transfer between applied and academic mathematics

Summary of the Report

The study investigates whether children’s arithmetic skills transfer between real-world applied mathematics and academic mathematics. The researchers surveyed 1,436 children working in markets in Kolkata and Delhi, India, and compared their performance to 471 school children with no market-selling experience. The key findings include:

1. Market-Working Children’s Math Skills
   • These children demonstrated strong arithmetic skills in real-world transactions, solving multi-step calculations quickly and accurately.
   • They effectively handled practical math problems related to market transactions, even when they involved complex or unfamiliar quantities.
   • Their skills were not based on memorization, stress reduction, or external help but were developed through daily practice in a meaningful context.
2. Academic Math Performance of Market-Working Children
   • Despite their real-world proficiency, these children struggled with abstract, school-based math problems.
   • They performed poorly on written arithmetic assessments, even for problems structurally simpler than those they solved in the market.
   • Their difficulty stemmed from the abstract nature of school math rather than literacy or cognitive limitations.
3. School-Going Children’s Performance
   • In contrast, children attending school without market experience performed well on simple, abstract arithmetic problems but struggled with applied, real-world problems.
   • They relied heavily on written calculations and inefficient problem-solving strategies, such as repeated addition instead of multiplication.
   • They often failed to apply school-taught math concepts in practical situations.
4. Limited Transfer of Math Skills in Both Directions
   • Neither group of children effectively transferred their skills from one domain (applied or academic) to the other.
   • Market-working children did not excel in abstract problems, and school-educated children failed to apply their knowledge to real-world tasks.
5. Implications for Education
   • The study highlights the need for educational reforms that bridge the gap between intuitive, real-world math and formal, abstract math taught in schools.
   • Existing curricula fail to leverage children’s natural numerical abilities and do not foster transferable mathematical thinking.

Implications for Math Education in Singapore’s Secondary Schools

Singapore’s secondary school math curriculum is globally recognized for its structured approach and strong emphasis on problem-solving. However, this study suggests areas for improvement to ensure students develop versatile and applicable math skills:

1. Enhancing Contextual Learning
   • While Singapore’s curriculum incorporates real-world problem-solving, more emphasis on practical applications in everyday scenarios could enhance retention and understanding.
   • Schools can integrate marketplace-style simulations, financial literacy tasks, and entrepreneurship projects into math lessons.
2. Bridging the Gap Between Conceptual and Applied Math
   • Teachers should explicitly connect abstract mathematical concepts to real-world contexts, using interdisciplinary projects that involve business calculations, engineering estimations, and data analysis.
   • Students could be tasked with practical applications like calculating costs for hypothetical start-ups or managing budgets for school events.
3. Reducing Over-Reliance on Formulaic Problem-Solving
   • Many Singaporean students excel at structured, step-by-step math problems but may struggle with open-ended, unstructured real-world applications.
   • Encouraging mental math and flexible problem-solving strategies, as seen in market-working children, could improve adaptability.
4. Developing Transferable Math Skills
   • The study highlights the failure of skill transfer between domains. Singapore’s math education could focus on cross-disciplinary learning, where students apply math concepts in physics, economics, and social studies.
   • Students should engage in hands-on projects that require them to analyze real-world problems, such as optimizing transportation routes or designing cost-effective solutions for community issues.
5. Encouraging Intuitive Number Sense
   • Instead of solely emphasizing procedural fluency, Singaporean schools can encourage estimation, mental calculation, and pattern recognition.
   • Using techniques like number decomposition (e.g., breaking 47 × 8 into (50 × 8) – (3 × 8)) can make calculations more intuitive and efficient.
6. Promoting Flexibility in Problem-Solving
   • Introducing different problem-solving approaches, including heuristic methods and algorithmic reasoning, could help students develop a deeper understanding of numbers beyond procedural rules.
   • Classroom activities could include puzzles, strategy games, and real-world mathematical modeling challenges.

Conclusion

Singapore’s math education system is already strong, but incorporating more real-world problem-solving, intuitive reasoning, and cross-context skill transfer could enhance students’ long-term math competence. By integrating applied learning methods inspired by the market-working children in this study, secondary school education can better prepare students for real-world mathematical challenges.

Minister Chan Chun Sing’s interview highlighted several key shifts in Singapore’s education system that directly affect students. These changes aim to reduce academic stress, promote diverse strengths, and encourage lifelong learning. Here’s how students are impacted:

1. More Personalized Learning Experiences

• The introduction of Full Subject-Based Banding (FSBB) allows students to take subjects at different difficulty levels based on their strengths.
  • Impact:
    • Students are no longer limited by their overall academic performance—they can excel in subjects they are strong in while receiving extra support in weaker subjects.
    • Encourages students to focus on their individual strengths instead of trying to be good at everything.
    • Reduces the stigma associated with weaker subject performance, fostering a more inclusive learning environment.

2. Shift in Academic Culture: Less Stress, More Holistic Development

• Policies like “Learn More, Test Less” and the revamp of the PSLE scoring system aim to reduce unnecessary academic pressure.
  • Impact:
    • Students are no longer judged on a fine-grained numerical ranking but on broader achievement bands, reducing unhealthy competition.
    • More focus on developing soft skills, creativity, and problem-solving instead of just rote memorization.
    • However, some students and parents may still feel pressure to excel in other ways, such as through Direct School Admission (DSA) or co-curricular activities (CCAs).

3. Greater Exposure to a Diverse Peer Group

• FSBB mixes students of different academic abilities in the same class for common subjects.
  • Impact:
    • Students interact with a more diverse range of peers, promoting mutual respect and reducing social stratification based on academic results.
    • Encourages a collaborative, inclusive mindset rather than a “top vs. bottom” mentality.

4. More Independent and Self-Directed Learning

• Schools are incorporating technology and AI-driven learning tools, similar to gamification in video games.
  • Impact:
    • Students have access to personalized, AI-driven resources that adapt to their individual learning pace.
    • Encourages self-directed learning, preparing students for university and lifelong learning.
    • However, students must develop good time management skills as some learning models involve self-paced study with gaps in their school schedules.

5. Changing Definition of Success

• There is a stronger push for students to discover and play to their strengths rather than comparing themselves to others.
  • Impact:
    • Students are encouraged to develop a growth mindset, understanding that different people have different talents.
    • Less focus on relative performance (i.e., “Am I better than my peers?”) and more on individual progress (i.e., “How can I improve myself?”).
    • However, shifting away from a grade-centric mindset is still a work in progress, and some students may struggle with the transition.

6. More Career-Focused and Real-World Learning Opportunities

• Universities and polytechnics are emphasizing lifelong learning and skills-based education.
  • Impact:
    • Students are better prepared for real-world job expectations, as universities focus on skills like problem-solving, collaboration, and innovation.
    • More opportunities for internships, hands-on learning, and career-relevant projects.
    • Less emphasis on pure academic results in university admissions and job hiring—employers are looking for well-rounded graduates.

7. Tuition May Become Less Necessary for Some, But Shift in Focus for Others

• With more personalized and accessible school resources, some students may rely less on tuition.
  • Impact:
    • Students who struggle with certain subjects can get school-based support rather than depending on external tuition.
    • For high-achieving students, tuition may shift towards enrichment rather than remedial help(e.g., leadership programs, coding courses, entrepreneurship training).
    • Parental pressure may still exist, pushing students to pursue additional tuition in areas like CCAs, DSA prep, and interview coaching instead of just academics.

Conclusion: A More Flexible, Student-Centered Education System

• Positive Impact:
  • Students can learn at their own pace and focus on their strengths.
  • Less stress from high-stakes exams and competition.
  • More diverse and inclusive learning environments.
  • Stronger emphasis on lifelong learning and career readiness.
• Challenges:
  • Cultural shifts take time—some students and parents still feel pressure to “stand out” in other ways.
  • Increased independence means students need self-discipline and time management skills.
  • Some students may struggle with adapting to less structured learning environments.

Final Thoughts

Overall, the changes in Singapore’s education system aim to create well-rounded, lifelong learners rather than just top scorers. However, students will need to adapt to a less rigid, more self-directed learning culture.

Minister Chan Chun Sing’s interview highlighted several key aspects of how Singapore’s evolving education policies impact the tuition industry. Below are the main points:

1. Reduced Academic Stress Could Affect Tuition Demand

• The shift towards full subject-based banding (FSBB) and broader assessment methods (beyond just exam scores) aims to reduce excessive academic stress.
• By discouraging unhealthy competition and over-reliance on grades, there may be less pressure on parents to enroll their children in tuition classes.
• However, parents may redirect their focus to non-academic differentiators like Direct School Admission (DSA), co-curricular activities (CCAs), and enrichment programs.

2. Tuition Industry Quickly Adapts to Policy Changes

• Tuition centers adjust their marketing strategies in response to MOE’s initiatives.
  • Example: When FSBB was introduced, tuition centers rebranded their services to help students cope with taking different subjects at different difficulty levels.
  • Similarly, after MOE reformed the Gifted Education Program (GEP), tuition centers expanded their preparatory courses to target multiple entry points, despite MOE’s intent to reduce stress.
• Minister Chan criticized tuition centers that exploit parental anxiety, using tactics like guilt-tripping parents into signing up their children for extra classes.

3. Technology & Mass Customization May Reduce the Need for Tuition

• MOE is leveraging AI, data analytics, and online learning resources to provide more personalized education.
• Examples include:
  • AI-powered essay feedback systems in schools.
  • Online personalized math exercises that adjust to students’ learning levels.
• These initiatives could reduce dependence on tuition for remedial or advanced learning.

4. Possible Regulation of Tuition Advertising & Practices

• Some tuition centers:
  • Pre-select top students and claim credit for their success.
  • Use misleading marketing tactics to suggest that tuition is essential for academic excellence.
• MOE is in discussions with advertising regulators to establish ethical guidelines for tuition industry marketing.

5. Private Tuition May Go Underground if Over-Regulated

• Some countries have banned excessive private tuition, but this has led to an underground tuition industrywhere only the wealthy can afford elite private tutors.
• Singapore is unlikely to ban tuition but aims to reduce over-reliance by improving public education accessibility.

Conclusion: Tuition Industry Will Adapt, But Its Role May Shift

• While tuition will not disappear, demand for traditional rote-learning-based tuition may decline.
• The industry might shift focus to:
  • Skills-based and enrichment programs.
  • Preparing students for non-academic pathways like DSA and CCAs.
  • Providing support for students with special needs or weak subjects.
• MOE’s success in promoting lifelong learning and reducing academic pressure will determine whether tuition remains a necessity or becomes a supplementary choice.

To understand principal values of inverse trigonometric functions, let’s break it down step by step:

1. Why Do We Need Inverse Trig Functions?

Trigonometric functions (like \( \sin \theta \), \( \cos \theta \), \( \tan \theta \)) take an angle and give a number. Inverse trig functions (like \(\sin^{-1} x\), \(\cos^{-1} x \), \(\tan^{-1} x \)) do the opposite: they take a number and return an angle.

2. The Problem: Periodicity

Trig functions are periodic, meaning they repeat their values. For example:

• \( \sin 30^\circ = \sin 150^\circ = 0.5 \)
• \( \cos 0^\circ = \cos 360^\circ = 1 \)

This means there are infinitely many angles that give the same trig value. But a function can only have one output for each input. So, how do we define inverse trig functions?

3. Solution: Restrict the Domain (Principal Values)

To make inverse trig functions work, we restrict their range (the angles they can output) to a specific interval called the principal value. This ensures each input gives exactly one angle.

Principal Value Ranges:

4. Key Takeaways

• Principal values are the “main” angles returned by inverse trig functions.
• Calculators use these ranges to give a single answer (e.g., typing \( \sin^{-1}(0.5) \) gives \( \frac{\pi}{6} \)).

Express \( \frac{(x+2)^2}{x^2(x-2)} \) as the sum of 3 partial fractions.

Take note that \( x^2 \) is a repeated factor

\( \begin{aligned} \frac{(x+2)^2}{x^2(x-2)}=\frac{x^2+4 x+4}{x^2(x-2)} \\ \text { Let } \frac{x^2+4 x+4}{x^2(x-2)} =\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-2} \\ \quad x^2+4 x+4 =A x(x-2)+B(x-2)+C x^2\end{aligned}\)

Find values of \( A \) and \( B \) by substituting suitable values of \( x \)

To find \( B \), sub \( x = 0 \)

To find \( C \), sub \( x = 2 \)

Now that we know \( B \) and \( C \), sub \( x = 1 \) to find \( A \)

(i) Find the value of \(a\) and of \(b\) for which \(2 x^2+3 x-2\) is a factor of \( 2 x^4+3 x^3+a\left(x^2+x\right)+b \)

(ii) Using the values of \(a\) and \(b\) found in part (i), solve the equation \(2 x^4+3 x^3+a\left(x^2+x\right)+b=0 \)

(i) If \(2 x^2+3 x-2\) is a factor of \( 2 x^4+3 x^3+a\left(x^2+x\right)+b \), the factors of \(2 x^2+3 x-2\) are also factors of \( 2 x^4+3 x^3+a\left(x^2+x\right)+b \)

Factorise \(2 x^2+3 x-2\) to get \( (2x-1)(x+2) \)

Apply factor theorem

Let \( f(x) = 2 x^4+3 x^3+a\left(x^2+x\right)+b \) ,

\( f(0.5) =0 \) , since \( (2x-1) \) is a factor

\( 2 + 3a +4b = 0 \)

\( f(-2) =0 \) , since \( (x+2) \) is a factor

\( 8 + 2a + b = 0 \)

Solving simultaneously, \(a = – 6 \), \( b = 4 \)

(ii) To solve \(2 x^4+3 x^3-6\left(x^2+x\right)+4=0 \)

Perform long division

The quotient of \( \left(2x^4 +3x^3 −6(x^2+x)+4 \right) \div \left(2x^2+3x−2 \right)\) is:\(x^2−2 \)

Hence \( \left(2x^2+3x−2 \right) \left(x^2−2 \right) = 0 \)