5.1 Q-7

Exercise 5.1 — Question 7

This exercise applies polynomial division, the Remainder Theorem, and the Factor Theorem to evaluate remainders and identify factors of polynomials.

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Key Concepts

Polynomial Division (SLO M-11-A-42)

To divide a polynomial $P(x)$ of degree up to 4 by a linear factor $(x-k)$, use synthetic division:

1. Write the coefficients of $P(x)$.
2. Use $k$ as the synthetic divisor.
3. Bring down the leading coefficient, multiply by $k$, add to the next coefficient, and repeat.
4. The final value is the remainder; the preceding values are coefficients of the quotient.

Remainder Theorem (SLO M-11-A-43)

Example: Find the remainder when $P(x)=2x^3+3x^2-4x+1$ is divided by $(x+2)$.

Here $a=-2$: $P(-2)=2(-2{)}^3+3(-2{)}^2-4(-2)+1=-16+12+8+1=5$

Remainder $=5$.

Factor Theorem (SLO M-11-A-44)

This is a special case of the Remainder Theorem where the remainder is zero.

Steps to factorize a cubic polynomial $P(x)$:

1. Find a value $a$ such that $P(a)=0$ (try $\pm 1,\pm 2,\dots$).
2. Divide $P(x)$ by $(x-a)$ using synthetic division to get a quadratic quotient $Q(x)$.
3. Factorize $Q(x)$ by inspection or the quadratic formula.
4. Write $P(x)=(x-a)\cdot Q(x)$.

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

• Test $x=1$: $P(1)=1-6+11-6=0$ ✓ → $(x-1)$ is a factor.
• Synthetic division gives quotient $x^2-5x+6$.
• $x^2-5x+6=(x-2)(x-3)$.
• Therefore: $P(x)=(x-1)(x-2)(x-3)$.

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