5.1 Q-2

Exercise 5.1 — Question 2

Key Concepts

This question applies polynomial long division and the Remainder Theorem to divide polynomials of degree up to 4 by linear or quadratic polynomials.

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Polynomial Division Algorithm

For any polynomial $P(x)$ divided by a non-zero polynomial $D(x)$:

$P(x)=D(x)\cdot Q(x)+R(x)$

where:

• $Q(x)$ = quotient
• $R(x)$ = remainder, with $\mathrm{deg}(R)<\mathrm{deg}(D)$

When dividing by a linear factor $(x-k)$, the remainder $R$ is a constant.

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Remainder Theorem

Proof sketch: $P(x)=(x-a)\cdot Q(x)+R$ Substituting $x=a$: $P(a)=(a-a)\cdot Q(a)+R=0+R=R$

So the remainder $R=P(a)$. No long division is required.

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Method: Long Division

To divide $P(x)$ by $D(x)$:

1. Arrange both polynomials in descending powers of $x$, inserting $0$ for missing terms.
2. Divide the leading term of $P(x)$ by the leading term of $D(x)$ to get the first term of $Q(x)$.
3. Multiply $D(x)$ by that term and subtract from $P(x)$.
4. Bring down the next term and repeat until the degree of the remainder is less than the degree of $D(x)$.

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Worked Example

Divide $P(x)=2x^3+3x^2-4x+1$ by $(x+2)$.

Using the Remainder Theorem (to find remainder only): $(x+2)=(x-(-2))⟹a=-2$ $P(-2)=2(-2{)}^3+3(-2{)}^2-4(-2)+1=-16+12+8+1=5$

Remainder = 5

Using Long Division (to find quotient and remainder):

$\text{require}enclose2x^3+3x^2-4x+1÷(x+2)$

$\therefore Q(x)=2x^2-x-2,R=5$

Verification: $(x+2)(2x^2-x-2)+5=2x^3+3x^2-4x+1$ ✓

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