Math Core

You've been combining like terms since the first unit. A polynomial is simply an expression built from those kinds of terms, and adding or subtracting polynomials is combining like terms on a larger scale. The vocabulary in this lesson (degree, standard form, leading coefficient) will follow you through the rest of the course.

Monomials and polynomials

Definition

Monomial

A monomial is a number, a variable, or a product of a number and variables with whole-number exponents. Examples: $7$ , $x$ , $-4x^3$ , $\tfrac{1}{2}ab^2$ .

A polynomial is a monomial or a sum of monomials. Each monomial is a term of the polynomial.

Some expressions look close but are not polynomials:

expression	polynomial?	reason
$3x^2 - 5x + 1$	yes	every exponent is a whole number
$\dfrac{2}{x} + 4$	no	$\dfrac{2}{x} = 2x^{-1}$ , a negative exponent
$\sqrt{x} + 1$	no	$\sqrt{x} = x^{1/2}$ , a fractional exponent
$\dfrac{x}{5} - 8$	yes	dividing by a number is fine: $\dfrac{1}{5}x - 8$

The rule is about the variable's exponents. A variable in a denominator or under a root is not allowed; fractions and roots of ordinary numbers, like $\tfrac{1}{5}$ or $\sqrt{2}$ , are fine as coefficients.

Degree, standard form and names

The degree of a term is the sum of the exponents on its variables. The degree of $-4x^3$ is $3$ ; the degree of $5x^2y$ is $2 + 1 = 3$ ; a nonzero constant like $9$ has degree $0$ .

Describing a polynomial

The degree of a polynomial is the greatest degree of any of its terms.
A polynomial is in standard form when its terms are written from highest degree to lowest.
In standard form, the first term is the leading term and its coefficient is the leading coefficient.

Take $6x - 2x^4 + 9 + x^2$ . Rearranged into standard form it becomes

-2x^4 + x^2 + 6x + 9.

Its degree is $4$ and its leading coefficient is $-2$ . Notice that the leading coefficient is not the first number you see in the original; you have to find the highest-degree term first.

Polynomials are also named by how many terms they have and by their degree:

number of terms	name	degree	name
1	monomial	0	constant
2	binomial	1	linear
3	trinomial	2	quadratic
4 or more	polynomial	3	cubic

So $x^2 - 9$ is a quadratic binomial, and $4x^3 + x - 7$ is a cubic trinomial.

Worked example: Naming a polynomial

Write $5 + 3x^2 - x^3$ in standard form, then give its degree, leading coefficient and name.

Order the terms by degree, highest first: $-x^3 + 3x^2 + 5$ .

Degree: $3$ (from $-x^3$ ).
Leading coefficient: $-1$ , since $-x^3$ means $-1 \cdot x^3$ .
It has three terms and degree $3$ , so it is a cubic trinomial.

Adding polynomials

To add polynomials, remove the parentheses and combine like terms. Parentheses preceded by a plus sign can simply be dropped, because adding a group is the same as adding each of its terms.

Worked example: Adding two polynomials

Find $(4x^2 - 3x + 8) + (x^2 + 7x - 10)$ .

\begin{aligned} &= 4x^2 - 3x + 8 + x^2 + 7x - 10 \\ &= (4x^2 + x^2) + (-3x + 7x) + (8 - 10) && \text{group like terms} \\ &= 5x^2 + 4x - 2 \end{aligned}

You can also line the polynomials up vertically, like adding multi-digit numbers. Put like terms in the same column, and leave a gap (or write a $0$ term) when a power is missing:

\begin{array}{r} 2x^3 + 0x^2 - 5x + 1 \\ +\;\; x^3 + 6x^2 + 2x - 4 \\ \hline 3x^3 + 6x^2 - 3x - 3 \end{array}

Subtracting polynomials

Subtracting a polynomial means subtracting every term in it. The minus sign in front of the parentheses acts like multiplying by $-1$ , so it flips the sign of each term inside. After that, it's an addition problem.

Worked example: Subtracting two polynomials

Find $(7y^2 + 2y - 5) - (3y^2 - 4y + 6)$ .

Distribute the $-1$ to the second polynomial, then combine:

\begin{aligned} &= 7y^2 + 2y - 5 - 3y^2 + 4y - 6 && \text{every sign in the second group flips} \\ &= (7y^2 - 3y^2) + (2y + 4y) + (-5 - 6) \\ &= 4y^2 + 6y - 11 \end{aligned}

Common mistake

The most common mistake is flipping only the first sign of the subtracted polynomial:

(7y^2 + 2y - 5) - (3y^2 - 4y + 6) \ne 7y^2 + 2y - 5 - 3y^2 - 4y + 6.

The subtraction applies to the whole group. Rewrite $-(3y^2 - 4y + 6)$ as $-3y^2 + 4y - 6$ before you combine anything.

Worked example: A missing polynomial

What polynomial must be added to $2x^2 - x + 3$ to get $5x^2 + 4x - 1$ ?

If $P + (2x^2 - x + 3) = 5x^2 + 4x - 1$ , then $P$ is the difference:

\begin{aligned} P &= (5x^2 + 4x - 1) - (2x^2 - x + 3) \\ &= 5x^2 + 4x - 1 - 2x^2 + x - 3 \\ &= 3x^2 + 5x - 4 \end{aligned}

Check: $(3x^2 + 5x - 4) + (2x^2 - x + 3) = 5x^2 + 4x - 1$ . ✓

Tip

Check a sum or difference by substituting a simple value such as $x = 1$ . In the last example, the original expressions give $(5 + 4 - 1) - (2 - 1 + 3) = 8 - 4 = 4$ , and the answer gives $3 + 5 - 4 = 4$ . Matching values don't prove the answer is right, but a mismatch always means there's an error.

Practice

Practice 1

What is the degree of $4x^3 - 7x^5 + 2x - 9$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

What is the leading coefficient of $3x - 8x^4 + x^2 + 6$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which name describes $5x^3 - 2x$ ?

Practice 4

Find $(5x^2 + 3x - 4) + (2x^2 - 7x + 9)$ .

Practice 5

Find $(6y^2 - 2y + 1) - (4y^2 + 5y - 3)$ .

Practice 6

Simplify $(3a^3 - a + 8) - (a^3 - 4a^2 - 6a + 10)$ . What is the coefficient of $a$ in the result?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

A triangle has sides of length $2x^2 + x$ , $x^2 - 3x + 4$ and $3x + 5$ . Which expression gives its perimeter?

Practice 8

What polynomial must be added to $x^2 - 4x + 7$ to get $3x^2 + x - 2$ ?