Lesson 5.1: Add and Subtract Polynomials

New Concept

This lesson introduces polynomials, expressions like $3x^2-5x+7$. You'll learn to classify them by degree and terms, then master adding and subtracting polynomials and polynomial functions.

What’s next

Next up, you'll work through interactive examples and practice cards to master adding, subtracting, and evaluating these new expressions.

Property

A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form $ax^m$, where $a$ is a constant and $m$ is a whole number.

Polynomials

• polynomial—A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.
• monomial—A polynomial with exactly one term is called a monomial.
• binomial—A polynomial with exactly two terms is called a binomial.
• trinomial—A polynomial with exactly three terms is called a trinomial.

Examples

• $5x^3 - 2x + 1$ is a trinomial because it has three terms.

Property

• The degree of a term is the sum of the exponents of its variables.
• The degree of a constant is $0$.
• The degree of a polynomial is the highest degree of all its terms.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial.

Examples

• The polynomial $2x^5 - 3x^2 + 7$ has a degree of $5$, which is the highest power of $x$.

Property

To add or subtract polynomials, combine like terms. Like terms are monomials that have the same variables with the same exponents. The Commutative Property allows rearranging terms to group like terms together. When subtracting polynomials, be careful to distribute the negative sign to every term in the polynomial being subtracted.

Examples

• To find the sum: $(3x^2 + 5x - 2) + (x^2 - 2x + 7) = (3x^2 + x^2) + (5x - 2x) + (-2 + 7) = 4x^2 + 3x + 5$.

• To find the difference: $(8a^2 - 4a + 1) - (3a^2 - a - 5) = 8a^2 - 4a + 1 - 3a^2 + a + 5 = 5a^2 - 3a + 6$.

Property

A polynomial function is a function whose range values are defined by a polynomial. To evaluate a polynomial function, substitute the given value for the variable and then simplify using the order of operations.

Examples

• For $f(x) = 2x^2 - 3x + 1$, find $f(3)$. Substitute $x=3$: $f(3) = 2(3)^2 - 3(3) + 1 = 2(9) - 9 + 1 = 10$.

• For $g(x) = x^3 + 5$, find $g(-2)$. Substitute $x=-2$: $g(-2) = (-2)^3 + 5 = -8 + 5 = -3$.

Property

For functions $f(x)$ and $g(x)$,

Examples

• Let $f(x) = 2x+3$ and $g(x) = x^2-1$. The sum is $(f+g)(x) = (2x+3) + (x^2-1) = x^2 + 2x + 2$.

• Let $f(x) = 4x^2 - 5$ and $g(x) = x^2 + 2x$. The difference is $(f-g)(x) = (4x^2 - 5) - (x^2 + 2x) = 3x^2 - 2x - 5$.