5.2 Q-3

Exercise 5.2 — Question 3

This question applies the Remainder Theorem and Factor Theorem to polynomials, typically requiring you to:

1. Find the remainder when a polynomial is divided by a linear factor.
2. Determine whether a given linear expression is a factor.
3. Fully factorize a cubic polynomial.

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Key Theorems

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 $P(a)=0$.

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Method: Factorizing a Cubic Polynomial

To factorize $P(x)=a{x}^3+b{x}^2+cx+d$:

Step 1 — Hit and Trial: Test factors of the constant term $d$ (divided by factors of leading coefficient $a$) as potential zeros. Try $x=\pm 1,\pm 2,\dots$ until $P(a)=0$.

Step 2 — Synthetic / Long Division: Once a zero $x=a$ is found, divide $P(x)$ by $(x-a)$ to obtain a quadratic quotient $Q(x)$.

Step 3 — Factorize the Quadratic: Factor $Q(x)$ by inspection, completing the square, or the quadratic formula.

Step 4 — Write the Full Factorization: $P(x)=(x-a)\cdot Q(x)$

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Worked Example

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

Step 1: Test $x=1$: $P(1)=1-6+11-6=0✓$ So $(x-1)$ is a factor.

Step 2: Divide $P(x)$ by $(x-1)$ using synthetic division: $x^3-6x^2+11x-6=(x-1)(x^2-5x+6)$

Step 3: Factorize $x^2-5x+6=(x-2)(x-3)$.

Step 4: $P(x)=(x-1)(x-2)(x-3)$

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Remainder Theorem Application

To find the remainder when $P(x)$ is divided by $(x-a)$, simply evaluate $P(a)$.

Example: Find the remainder when $P(x)=x^3-4x^2+3x-2$ is divided by $(x-3)$: $P(3)=27-36+9-2=-2$ Remainder $=-2$.

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