5.2 Q-1

Exercise 5.2 — Polynomial Division and Remainder Theorem

This exercise covers dividing polynomials (degree up to 4) by linear or quadratic polynomials, and applying the Remainder Theorem.

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Key Concepts

Polynomial Long Division

To divide $P(x)$ by $D(x)$, we find quotient $Q(x)$ and remainder $R(x)$ such that:

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

where $\mathrm{deg}(R)<\mathrm{deg}(D)$.

Steps:

1. Divide the leading term of $P(x)$ by the leading term of $D(x)$.
2. Multiply the result by $D(x)$ and subtract from $P(x)$.
3. Repeat with the new polynomial until the degree of the remainder is less than the degree of $D(x)$.

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Synthetic Division (for linear divisors)

When dividing by $(x-k)$, synthetic division provides a shortcut:

1. Write the coefficients of $P(x)$ in a row (use $0$ for missing terms).
2. Bring down the first coefficient.
3. Multiply by $k$, add to the next coefficient, and repeat.
4. The last number is the remainder; the others are coefficients of $Q(x)$.

Example: Divide $P(x)=x^3-4x^2+5x-2$ by $(x-2)$:

$\begin{array}{rrrrr}2& 1& -4& 5& -2\\ & & 2& -4& 2\\ & 1& -2& 1& 0\end{array}$

Quotient: $x^2-2x+1$, Remainder: $0$.

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Remainder Theorem

Proof sketch: By the division algorithm, $P(x)=(x-a)Q(x)+R$. Setting $x=a$ gives $P(a)=R$.

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

$P(2)=8-12+4=0$

Remainder $=0$, so $(x-2)$ is actually a factor.

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Worked Questions

The questions in Exercise 5.2 ask you to:

• Perform polynomial long division and state the quotient and remainder.
• Use the Remainder Theorem to find remainders without full division.
• Verify results by checking $P(a)=$ remainder.