5.2 Factorization and Remainder Theorem

Polynomial Division

A polynomial $P(x)$ of degree $n$ can be divided by another polynomial $D(x)$ of degree $m\le n$ to give a quotient $Q(x)$ and a remainder $R(x)$:

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

where the degree of $R(x)$ is less than the degree of $D(x)$.

Long Division

For dividing by a quadratic or higher-degree polynomial, use polynomial long division — the same algorithm as numerical long division.

Example: Divide $2x^3-3x^2+x-5$ by $x^2-1$.

$2x^3-3x^2+x-5=(x^2-1)(2x-3)+(3x-8)$

Quotient: $2x-3$, Remainder: $3x-8$.

Synthetic Division

For dividing by a linear factor $(x-k)$, synthetic division is faster:

1. Write the coefficients of $P(x)$ in a row (use $0$ for missing terms).
2. Write $k$ to the left.
3. Bring down the leading coefficient.
4. Multiply by $k$, write under the next coefficient, add.
5. Repeat until done. The last number is the remainder; the rest are coefficients of the quotient.

Example: Divide $2x^3-15x^2+16x+12$ by $(x-2)$:

$\begin{array}{rrrrr}2& 2& -15& 16& 12\\ & & 4& -22& -12\\ & 2& -11& -6& 0\end{array}$

Quotient: $2x^2-11x-6$, Remainder: $0$.

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Remainder Theorem

This means you can find the remainder without performing division — just 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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Factor Theorem

This is a special case of the Remainder Theorem: if the remainder $P(a)=0$, then $(x-a)$ divides $P(x)$ exactly.

For a zero at $x=\frac{p}{q}$, the corresponding integer-coefficient factor is $(qx-p)$.

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Factorizing a Cubic Polynomial

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

Step 1 — Hit and Trial

Test factors of $d$ (and $\frac{p}{q}$ where $q$ divides $a$) as potential zeros. By the Rational Root Theorem, any rational zero $\frac{p}{q}$ must satisfy:

• $p$ divides the constant term $d$
• $q$ divides the leading coefficient $a$

Step 2 — Synthetic Division

Once a zero $x=a$ is found, divide $P(x)$ by $(x-a)$ using synthetic division to obtain a quadratic $Q(x)$.

Step 3 — Factor the Quadratic

Factor $Q(x)$ by mid-term breaking or the quadratic formula:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

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

• Test $x=1$: $P(1)=1-6+11-6=0$ ✓
• Divide by $(x-1)$: quotient is $x^2-5x+6$
• Factor: $x^2-5x+6=(x-2)(x-3)$
• Result: $P(x)=(x-1)(x-2)(x-3)$

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Real-World Applications

The Remainder and Factor Theorems have important applications:

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Summary