Polynomial division and factor theorems

What you’ll learn

• Divide a polynomial by a linear factor.
• Use factors and remainders to solve polynomial problems.

Understand the idea

Polynomial division separates a polynomial into a divisor times a quotient plus a remainder. When the divisor is linear, the remainder is constant. Keep missing powers in the working with coefficient zero so that like terms stay aligned.

For division by $x-a$, substituting $x=a$ makes the divisor vanish, leaving the remainder $f(a)$. A zero remainder makes $x-a$ a factor. For division by $px-q$, use $x=q/p$ when applying the same reasoning. Once one cubic factor is found, divide and solve the remaining quadratic.

Key formulae

$$f(x)=(x-a)Q(x)+f(a)$$

$$f(a)=0\iff (x-a)\text{ is a factor of }f(x)$$

$$\text{remainder on division by }px-q=f(q/p)$$

Worked example

Factorise $f(x)=x^3-4x^2-x+4$ completely.

1. Evaluate $f(1)=1-4-1+4=0$, so $x-1$ is a factor.
2. Divide by $x-1$ to obtain $x^2-3x-4$.
3. Factorise the quotient: $x^2-3x-4=(x-4)(x+1)$.

Answer: $f(x)=(x-1)(x-4)(x+1)$.