Polynomials in one variable

Expressions with powers of x

Look at these expressions: $3x + 5$, $x^2 - 7x + 2$, $4y^3 + y$. Each is a sum of terms. Each term is a number times a power of a variable. These are examples of polynomials.

Polynomials are everywhere. The area of a square of side $x$ is $x^2$. The area of a rectangle with sides $x + 2$ and $x + 3$ is $x^2 + 5x + 6$. The height of a ball thrown up is a polynomial in the time. Engineers use polynomials to draw smooth curves, and computers use them to calculate other functions. So it is worth knowing them well.

A polynomial in one variable $x$ has the form

where $n$ is a whole number $(0, 1, 2, \ldots)$ and the numbers $a_0, a_1, \ldots, a_n$ are constants. The constants are the coefficients of the terms. The last one, $a_0$, is the constant term.

Example: in $5x^3 - 2x^2 + x - 7$, the coefficient of $x^3$ is $5$, of $x^2$ is $-2$, of $x$ is $1$, and the constant term is $-7$. The sign belongs to the term, so the coefficient of $x^2$ is $-2$ and not $2$.

What is not a polynomial?

The power of $x$ must be a whole number.

• $\sqrt{x} + 2 = x^{1/2} + 2$ is not a polynomial. The power is $\dfrac{1}{2}$.
• $x + \dfrac{1}{x} = x + x^{-1}$ is not a polynomial. The power is $-1$.
• $x^{3/2} - 4$ is not a polynomial.

A good rule is: the variable never appears under a root sign, and never in a denominator.

The degree

The degree of a polynomial in one variable is the highest power of the variable that has a non-zero coefficient.

Look at $3x - 4x^5 + 2$. The highest power is not written first. The degree is $5$. It is a good habit to write a polynomial in standard form, with the powers going down: $-4x^5 + 3x + 2$.

A polynomial with one term is a monomial. With two terms, it is a binomial. With three, it is a trinomial. For example $7x - 5$ is a linear binomial, and $2x^2 - x + 1$ is a quadratic trinomial.

The number $0$ is the zero polynomial. It has no degree. A non-zero constant such as $9$ has degree $0$, because $9 = 9x^0$.

Set $a = 0$. The curve turns into a parabola, and the degree drops to $2$. Set $b = 0$ as well, and it becomes a straight line.

The value of a polynomial

We often write a polynomial as $p(x)$. The symbol $p(2)$ means the value of the polynomial when $x = 2$. For $p(x) = x^2 - 3x + 4$:

Adding and multiplying

The sum and the difference of two polynomials are polynomials. We add the terms with the same power.

The product of two polynomials is also a polynomial. We multiply each term by each term.

The degree of a product is the sum of the degrees: $1 + 1 = 2$. The degree of a sum is at most the larger degree. It can be smaller: $(x^2 + x) + (-x^2 + 3) = x + 3$, where the $x^2$ terms cancel.

Practise with new numbers

Try it in Python

This program stores a polynomial as a list of coefficients, and it finds its degree and some values.

Common mistakes

• Taking the first power written as the degree. Look for the highest power.
• Forgetting the sign of the coefficient. In $x^2 - 7x$, the coefficient of $x$ is $-7$.
• Calling $\dfrac{1}{x}$ a polynomial. Its power is $-1$.
• Giving degree $1$ to a constant. A non-zero constant has degree $0$.

Remember these

• A polynomial in $x$ is a sum of terms, each with a coefficient times $x^n$, where $n$ is a whole number.
• Powers such as $x^{1/2}$ and $x^{-1}$ are not allowed.
• The degree is the highest power. Degree $1$ is linear, $2$ is quadratic and $3$ is cubic.
• The degree of a product is the sum of the degrees.

Check yourself

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

1. What is the degree of $6x^4 - x^2 + 3$?

2. Is $x^2 + \dfrac{3}{x}$ a polynomial?

3. What is the coefficient of $x$ in $4x^3 - 9x + 1$?

4. Find $p(-2)$ for $p(x) = x^3 + 2x - 1$.