Polynomials and Their Zeroes: Quick Revision

A brief note on Polynomials and Their Zeroes. Read it before a test. Try each question at the end before you open its answer.

Polynomials

• A polynomial in $x$ has only whole-number powers of $x$: $3x^4 - x + \sqrt{2}$ is one, $x + \dfrac{1}{x}$ is not.
• The degree is the highest power. Degree $1$, $2$ and $3$ are linear, quadratic and cubic.
• The value $p(a)$ comes from putting $x = a$.

Zeroes

• A zero is a number $a$ with $p(a) = 0$. On the graph, it is where the curve meets the x-axis.
• A polynomial of degree $n$ has at most $n$ zeroes.
• The zero of $ax + b$ is $-\dfrac{b}{a}$.

Two theorems

For a divisor $ax - b$, use $p\left(\dfrac{b}{a}\right)$.

A worked example

Find the remainder when $2x^3 - x^2 + 4$ is divided by $x - 2$, and by $2x - 1$.

1. By $x - 2$. $p(2) = 16 - 4 + 4 = 16$.
2. By $2x - 1$. It is zero at $x = \tfrac{1}{2}$: $p\left(\tfrac{1}{2}\right) = \tfrac{1}{4} - \tfrac{1}{4} + 4 = 4$.

Practice with new numbers

A program to try

The program works out remainders with the remainder theorem.

Mistakes to avoid

• Using the wrong sign. For $x + 3$, put $x = -3$.
• Counting a constant as degree 1. A constant such as $7$ has degree $0$.
• Expecting too many zeroes. A cubic has at most three.
• Calling $\sqrt{x}$ a polynomial term. Its power is $\tfrac{1}{2}$.

Check yourself

1. What is the degree of $7x^2 - 3x^5 + 1$?

2. What is the zero of $8x - 2$?

3. What is the remainder when $x^3 + x^2 + x + 1$ is divided by $x - 2$?

4. Is $x - 1$ a factor of $x^5 - 1$?