Polynomials and Rational Functions
4.2 Polynomial Division
Why Divide Polynomials
Division helps simplify rational expressions and test potential factors.
Definition: Division algorithm for polynomials
For polynomials $P(x)$ and nonzero $D(x)$:
$$P(x)=D(x)Q(x)+R(x),$$
where degree of $R$ is less than degree of $D$.
Example: Long division
Divide
$$x^3+2x^2-5x+6$$
by
$$x+3.$$
Show Solution
Use long division steps.
- $x^3\div x=x^2$, put $x^2$ in quotient.
- Multiply back: $x^2(x+3)=x^3+3x^2$.
- Subtract: $(x^3+2x^2)-(x^3+3x^2)=-x^2$.
- Bring down $-5x$.
- $-x^2\div x=-x$, place $-x$.
- Multiply back: $-x(x+3)=-x^2-3x$.
- Subtract: $(-x^2-5x)-(-x^2-3x)=-2x$.
- Bring down $+6$.
- $-2x\div x=-2$, place $-2$.
- Multiply back: $-2(x+3)=-2x-6$.
- Subtract: remainder $12$.
So
$$\frac{x^3+2x^2-5x+6}{x+3}=x^2-x-2+\frac{12}{x+3}. $$
MyOpenMath: Practice: polynomial long division (placeholder)
Synthetic Division
When dividing by $x-c$, synthetic division is faster.
Example: Synthetic division
Divide
$$2x^3-3x^2+4x-5$$
by
$$x-2.$$
Show Solution
Use $c=2$ with coefficients $2,-3,4,-5$.
Bring down 2.
- $2\cdot2=4$, add to $-3$ gives $1$.
- $1\cdot2=2$, add to $4$ gives $6$.
- $6\cdot2=12$, add to $-5$ gives $7$.
Quotient coefficients are $2,1,6$ with remainder $7$:
$$2x^2+x+6+\frac{7}{x-2}. $$