Zeros of a Polynomial

If you ask "where does this polynomial vanish?" you are asking for its zeros. Zeros are the most important numbers attached to a polynomial , they reveal its factors, they are the $x$-intercepts of its graph, and they are the solutions to the equation $p(x) = 0$. This lesson sets up the precise definition and what to expect.

Definitions

A zero (or root) of a polynomial $p(x)$ is any real number $c$ for which $p(c) = 0.$ A linear polynomial $p(x) = ax + b$ (with $a \ne 0$) has exactly one zero, namely $x = -\dfrac{b}{a}$. A quadratic polynomial can have $0, 1$ or $2$ zeros (in the real numbers). A polynomial of degree $n$ has at most $n$ real zeros.

The constant polynomial $p(x) = c$ with $c \ne 0$ has no zeros; it never equals zero. The zero polynomial $p(x) = 0$ is everywhere zero, so every real number is a "zero" of it , which is why we don't assign it a degree.

Concept and the basics

Linear case. To find the zero of $p(x) = ax + b$, set $ax + b = 0$ and solve: $x = -\dfrac{b}{a}$. Example: the zero of $2x - 6$ is $x = 3$ (because $2 \cdot 3 - 6 = 0$).

Quadratic case , by inspection. Sometimes you can guess. The polynomial $p(x) = x^2 - 5x + 6$ factors as $(x - 2)(x - 3)$, so its zeros are $2$ and $3$. The polynomial $p(x) = x^2 - 4$ factors as $(x-2)(x+2)$, so its zeros are $2$ and $-2$. We will see a general way to factorise quadratics later in the chapter.

Quadratic case , when factoring fails. Some quadratics have no real zeros at all. Take $p(x) = x^2 + 1$. For any real $x$, $x^2 \ge 0$, so $x^2 + 1 \ge 1 > 0$. The polynomial is never zero. A real-valued polynomial may simply have no real zero.

Cubic and higher. A cubic always has at least one real zero (because the graph crosses the $x$-axis at least once , it goes from $-\infty$ to $+\infty$). We will use the Factor Theorem to find such zeros in later lessons. A general degree-$n$ polynomial has at most $n$ real zeros.

Counting zeros. A polynomial $p(x)$ of degree $n$ has at most $n$ real zeros. (If we count complex zeros and "multiplicities", the count is exactly $n$. We will return to this in Class X.) So for example, a cubic has at most three real zeros , never four, never five.

Geometric picture. Graph $y = p(x)$. The zeros are exactly the $x$-coordinates where the curve crosses (or touches) the $x$-axis. A quadratic with no real zeros has a graph that sits entirely above or entirely below the axis. A cubic always crosses the axis at least once.

Finding the zero of $p(x) = ax + b$ in one line. Set $p(x) = 0$ and isolate $x$: $ax + b = 0 \implies x = -\frac{b}{a}.$

Using zeros backwards. If you are told a polynomial has certain zeros, you can write down a polynomial that does. A polynomial with zeros $2$ and $-3$ is, for example, $(x-2)(x+3) = x^2 + x - 6$.

Worked examples

Example 1. Find the zero of $p(x) = 3x - 12$.

Set $3x - 12 = 0$. Then $x = 4$.

Example 2. Find the zeros of $p(x) = x^2 - 9$.

Factor: $(x-3)(x+3)$. Zeros: $3$ and $-3$.

Example 3. Show that $-1$ is a zero of $p(x) = x^3 + 2x^2 - x - 2$.

Compute $p(-1) = (-1)^3 + 2(-1)^2 - (-1) - 2 = -1 + 2 + 1 - 2 = 0$. Yes, $-1$ is a zero.

Example 4. Show that $p(x) = x^2 + 4$ has no real zero.

For any real $x$, $x^2 \ge 0$, so $x^2 + 4 \ge 4 > 0$. The polynomial is never $0$.

Example 5. Write a quadratic polynomial whose zeros are $\dfrac{1}{2}$ and $-3$.

A quadratic with these zeros is $(x - \tfrac{1}{2})(x + 3)$. Expand: $x^2 + 3x - \tfrac{1}{2}x - \tfrac{3}{2} = x^2 + \tfrac{5}{2}x - \tfrac{3}{2}$. Clearing fractions: $2x^2 + 5x - 3$.

Try it yourself

1. Find the zero of $p(x) = 5x + 15$.
2. Find the zeros of $p(x) = x^2 - 16$.
3. Find the zeros of $p(x) = x^2 - 7x + 10$.
4. Verify that $2$ is a zero of $p(x) = x^3 - 4x^2 + x + 6$.
5. Show that $p(x) = x^2 + 9$ has no real zeros.
6. Write a polynomial of degree $3$ with zeros $0, 1, 2$.
7. Find a quadratic with zeros $4$ and $-5$.
8. Is $-2$ a zero of $p(x) = x^4 - 16$?
9. Find the zero of $p(x) = ax + b$ assuming $a \ne 0$.
10. Find all real zeros of $p(x) = x(x-1)(x+5)$.

Pitfalls / Insight

• A zero is a number, not a point on a graph. It is the $x$-value where $p(x) = 0$.
• A polynomial of degree $n$ has at most $n$ real zeros, not exactly $n$. The polynomial $x^2 + 1$ has $0$ real zeros.
• "Zero of the polynomial" and "zero of the equation" mean the same thing. Both ask for $x$ with $p(x) = 0$.

Insight. The most powerful question you can ask about a polynomial is "where does it vanish?" That single question , answered , gives the factorisation, the graph, and the solutions of the corresponding equation, all at once. The next lessons turn this question into a method.