5.2 Q-2

Exercise 5.2 — Question 2

This question applies polynomial division, the Remainder Theorem, and the Factor Theorem to factorize cubic polynomials.

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Key Concepts Recap

Remainder Theorem: When a polynomial $P(x)$ is divided by $(x-a)$, the remainder is $P(a)$.

Factor Theorem: $(x-a)$ is a factor of $P(x)$ if and only if $P(a)=0$.

Hit and Trial Method: To find a zero of $P(x)=a{x}^3+b{x}^2+cx+d$, test factors of the constant term $d$ (divided by factors of the leading coefficient $a$). Once a zero $x=k$ is found, divide $P(x)$ by $(x-k)$ to obtain a quadratic, then factorize the quadratic.

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Worked Example

Factorize $P(x)=x^3-6x^2+11x-6$.

Step 1 — Find a zero by hit and trial.

Factors of $-6$: $\pm 1,\pm 2,\pm 3,\pm 6$.

$P(1)=1-6+11-6=0✓$

So $(x-1)$ is a factor.

Step 2 — Divide $P(x)$ by $(x-1)$ using synthetic division.

$\begin{array}{rrrrr}1& 1& -6& 11& -6\\ & & 1& -5& 6\\ & 1& -5& 6& 0\end{array}$

Quotient: $x^2-5x+6$, Remainder: $0$.

Step 3 — Factorize the quotient.

$x^2-5x+6=(x-2)(x-3)$

Step 4 — Write the complete factorization.

$P(x)=(x-1)(x-2)(x-3)$

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General Procedure for Factorizing a Cubic

1. List all factors of the constant term.
2. Test each using the Factor Theorem until $P(a)=0$.
3. Perform synthetic (or long) division by $(x-a)$ to get a quadratic.
4. Factorize the quadratic by inspection, completing the square, or the quadratic formula.
5. Write the full factored form.

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