Polynomials, Division and the Factor Theorem

Starting point: what you already know

• A quadratic $a{x}^2+bx+c$ has a highest power of 2.
• Expanding: $(x-1)(x+3)=x^2+2x-3$.
• Factorising a quadratic and solving $a{x}^2+bx+c=0$.
• Substitution: for $P(x)=x^2+1$, $P(2)=5$.
• Long division of numbers: $47÷5=9$ remainder 2, so $47=5\times 9+2$.

The language of polynomials

A polynomial is a sum of terms with whole number powers of $x$:

$$P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$$

A root or zero of $P$ is a number $\alpha$ with $P(\alpha )=0$. On the graph, it is where the curve meets the x-axis.

Expressions like $\sqrt{x}$ or $\frac{1}{x}$ are not polynomials, because the powers are not whole numbers.

Dividing polynomials

Just like $47=5\times 9+2$, any polynomial can be divided by another:

$$P(x)=A(x)\times Q(x)+R(x)$$

where $Q(x)$ is the quotient and $R(x)$ is the remainder. The remainder has a lower degree than the divisor $A(x)$.

The remainder theorem

Put $x=a$ into $P(x)=(x-a)Q(x)+R$. The first part becomes zero, so:

Check the example: $P(1)=1-2+4-5=-2$. Same remainder, no division needed.

The factor theorem

If the remainder is zero, $(x-a)$ divides exactly.

Quick summary

• A polynomial has whole number powers. The degree is the highest power.
• $P(x)=A(x)Q(x)+R(x)$, with the remainder of lower degree than $A(x)$.
• Remainder theorem: dividing by $(x-a)$ leaves remainder $P(a)$.
• Factor theorem: $(x-a)$ is a factor exactly when $P(a)=0$.
• To factorise a cubic, test factors of the constant term, then divide.

Quiz