5.3 Factorization and Polynomial Division

Polynomial Division

A polynomial $P(x)$ of degree $n$ can be divided by a divisor $D(x)$ of degree less than or equal to $n$ to produce 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 strictly less than the degree of $D(x)$.

Long Division of Polynomials

To divide $P(x)$ by a linear factor $(x-a)$ or a quadratic factor:

1. Arrange both polynomials in descending powers of $x$, inserting $0$ for missing terms.
2. Divide the leading term of the dividend by the leading term of the divisor.
3. Multiply the result by the entire divisor and subtract.
4. Repeat until the degree of the remainder is less than the degree of the divisor.

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

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

So the quotient is $2x^2+x+3$ and the remainder is $1$.

Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear factor $(x-a)$:

• Write only the coefficients of $P(x)$.
• Use $a$ (the root of the divisor) as the synthetic divisor.
• Bring down, multiply, add — repeat.

Example: Divide $x^3-6x^2+11x-6$ by $(x-1)$ using synthetic division:

Quotient: $x^2-5x+6$, Remainder: $0$.

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Remainder Theorem

$P(x)=(x-a)\cdot Q(x)+P(a)$

Example: Find the remainder when $P(x)=x^3-4x+6$ is divided by $(x-2)$.

$P(2)=(2{)}^3-4(2)+6=8-8+6=6$

The remainder is $6$.

Extension: When dividing by $(ax-b)$, the remainder is $P\left(\frac{b}{a}\right)$.

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Factor Theorem

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

Factorizing a Cubic Polynomial

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

1. Find a root by testing factors of $\frac{d}{a}$ (Rational Root Theorem).
2. Divide $P(x)$ by $(x-a)$ to get a quadratic quotient $Q(x)$.
3. Factor $Q(x)$ by inspection, completing the square, or the quadratic formula.

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.
• Divide: $P(x)=(x-1)(x^2-5x+6)$.
• Factor the quadratic: $x^2-5x+6=(x-2)(x-3)$.
• Result: $P(x)=(x-1)(x-2)(x-3)$.

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Real-World Applications

Finding Dimensions from Volume

If the volume of a rectangular solid is given as a polynomial $V(x)$ and one dimension is a known factor $(x-a)$, divide $V(x)$ by $(x-a)$ to find the product of the remaining two dimensions, then factor the resulting quadratic.

Example: $V(y)=y^3-2y^2-y+2$ with one side $(y-2)$.

• $P(2)=8-8-2+2=0$ → $(y-2)$ is a factor.
• Divide: $V(y)=(y-2)(y^2-1)=(y-2)(y-1)(y+1)$.
• Dimensions: $(y-2)$, $(y-1)$, $(y+1)$.

Finding Area and Length

If the area of a rectangle is $A(x)$ and the width is $W(x)$, then: $L(x)=\frac{A(x)}{W(x)}$ found by polynomial division.

Other Applications

• Polynomial regression: Remainder and Factor Theorems help evaluate and simplify regression polynomials.
• Signal processing: Polynomial factorization is used in filter design.
• Coding theory: Cyclic codes use polynomial division over finite fields.

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