Zeroes of a polynomial

The value of a polynomial

If $p(x) = 2x^2 - 3x + 5$, then $p(2)$ means that we put $x = 2$ in the polynomial.

This number, $7$, is the value of $p(x)$ at $x = 2$.

Zeroes

A number $a$ is a zero of the polynomial $p(x)$ if $p(a) = 0$.

Example: $p(x) = x^2 - 3x + 2$. Then $p(1) = 1 - 3 + 2 = 0$ and $p(2) = 4 - 6 + 2 = 0$. So $1$ and $2$ are zeroes of $p(x)$. Also $p(0) = 2$, so $0$ is not a zero.

Finding a zero of $p(x)$ is the same as solving the equation $p(x) = 0$. So a zero of a polynomial is also called a root of the equation.

The zero of a linear polynomial

For $p(x) = ax + b$, with $a \ne 0$, we solve $ax + b = 0$.

Example: $p(x) = 3x - 6$ has the zero $x = \dfrac{6}{3} = 2$. The polynomial $2x - 7$ has the zero $\dfrac{7}{2}$.

A different kind of problem. For which value of $k$ is $2$ a zero of $x^2 - kx + 6$? Put $x = 2$: $4 - 2k + 6 = 0$, so $10 = 2k$ and $k = 5$. Check: $x^2 - 5x + 6$ at $x = 2$ gives $4 - 10 + 6 = 0$. ✓

How many zeroes?

• A non-zero constant, such as $p(x) = 5$, has no zero, because it is never $0$.
• A linear polynomial has exactly one zero.
• A polynomial of degree $n$ has at most $n$ zeroes.

A quadratic can have two zeroes, one zero or none (among real numbers). For example, $x^2 - 4$ has the zeroes $2$ and $-2$. The polynomial $x^2$ has only $0$. And $x^2 + 1$ has no real zero, because $x^2$ is never negative, so $x^2 + 1$ is always at least $1$.

Zeroes on a graph

The graph of $y = p(x)$ meets the $x$-axis where $y = 0$. So the zeroes of $p(x)$ are the $x$-coordinates of the points where the graph meets the $x$-axis.

The quadratic has two zeroes, and the cubic has three. A polynomial of degree $n$ can cross the axis at most $n$ times. Some polynomials, such as $x^2 + 1$, never cross the axis, and they have no real zero.

A graph helps in two ways. First, we can see how many zeroes there are. Second, we can read a rough value, and then check it by putting it in the polynomial.

Raise the lift slowly. The two zeroes move towards each other, meet at one point, and then disappear: the curve no longer touches the x-axis.

Finding whole number zeroes

For a polynomial with whole number coefficients, any whole number zero must divide the constant term. For $x^3 - 2x^2 - 5x + 6$, the constant is $6$. So the only whole numbers to try are $\pm 1, \pm 2, \pm 3, \pm 6$.

• $p(1) = 1 - 2 - 5 + 6 = 0$. So $1$ is a zero.
• $p(-2) = -8 - 8 + 10 + 6 = 0$. So $-2$ is a zero.
• $p(3) = 27 - 18 - 15 + 6 = 0$. So $3$ is a zero.

That is all three zeroes. The degree is $3$, so there can be no more.

Practise with new numbers

Try it in Python

This program tries the whole numbers from $-5$ to $5$ in a cubic polynomial, and it marks the zeroes.

Common mistakes

• Confusing the value with the zero. The value $p(a)$ can be any number. The number $a$ is a zero only if the value is $0$.
• Forgetting the sign in $-\dfrac{b}{a}$. The zero of $3x + 6$ is $-2$.
• Saying that every quadratic has two zeroes. It can have two, one or none.
• Saying that $0$ is always a zero. It is a zero only if the constant term is $0$.

Remember these

• $a$ is a zero of $p(x)$ if $p(a) = 0$.
• The zero of $ax + b$ is $-\dfrac{b}{a}$.
• A polynomial of degree $n$ has at most $n$ zeroes. They are where its graph meets the $x$-axis.
• A whole number zero divides the constant term.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the value of $p(x) = x^2 - 4x + 1$ at $x = 3$.

2. Find the zero of $5x + 10$.

3. Is $2$ a zero of $x^2 - 5x + 6$?

4. For which value of $k$ is $1$ a zero of $x^2 + kx - 4$?