Polynomials, Cubics, Partial Fractions practice questions

Secondary 3 topics from the Additional Mathematics 4049 syllabus. Every question below has a full worked solution.

When a student gets a question wrong in Math Amigo, it does not show the answer straight away. It reads the working, identifies the step that went wrong, and asks a question that points at the correction. The worked solution is there once a genuine attempt has been made.

Practise polynomials, cubics, partial fractions free

What this topic covers

Worked examples

Example 1

Divide 3x3−9x2+10x−103x^{3} - 9x^{2} + 10x - 10 by (x−2)(x - 2). State the quotient and remainder.
Show the worked solution
  • $\text{Perform long division of } 3x^{3} - 9x^{2} + 10x - 10 \text{ by } (x - 2)$
  • $\text{Quotient} = 3x^{2} - 3x + 4$
  • $\text{Remainder} = -2$
  • $\therefore 3x^{3} - 9x^{2} + 10x - 10 = (x - 2)(3x^{2} - 3x + 4) - 2$

Answer: "q2":"3","q1":"−3","q0":"4","remainder":"−2"{"q2":"3","q1":"-3","q0":"4","remainder":"-2"}

Example 2

Given that (x−3)(x - 3) is a factor of P(x)=x3+kx−18P(x) = x^3 + kx - 18, find kk.
Show the worked solution
  • $\text{Since } (x - 3) \text{ is a factor, } P(3) = 0 \quad \scriptsize\textit{Factor Theorem}$
  • $(3)^3 + k(3) - 18 = 0$
  • $3k = -9$
  • $k = -3$

Answer: −3-3

Example 3

Solve x3−4x2−4x+16=0x^{3} - 4x^{2} - 4x + 16 = 0.
Show the worked solution
  • $\text{Let } f(x) = x^{3} - 4x^{2} - 4x + 16$
  • $\text{Try } f(2) = 0 \quad \scriptsize\textit{Factor Theorem}$
  • $\therefore (x - 2) \text{ is a factor}$
  • $\text{Divide to find the quadratic factor, then factorise}$
  • $x = -2, \; 2, \; 4$

Answer: −2,2,4-2, 2, 4

Questions parents ask

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Which syllabus does this follow?

Additional Mathematics (4049), for Singapore secondary students. Math Amigo covers Secondary 1 to 4, both O-Level Mathematics and Additional Mathematics.

Will it just show my child the answer?

No. On a wrong answer it looks at the working, points to the step that went wrong, and asks a question that helps the student find the correction. The full worked solution is available after a genuine attempt.

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