9-1 Adding and Subtracting Polynomials

Property

A monomial is a term of the form $ax^m$, where $a$ is a constant and $m$ is a whole number.
A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.
A polynomial with exactly one term is called a monomial.
A polynomial with exactly two terms is called a binomial.
A polynomial with exactly three terms is called a trinomial.

Examples

• $15x^4$ is a monomial because it has one term.
• $y^2 - 25$ is a binomial because it has two terms.
• $3a^2 - 6a + 9$ is a trinomial because it has three terms.

Explanation

Think of polynomials as a family. Monomials (one term), binomials (two terms), and trinomials (three terms) are specific members. We use these names for them, and call everything else with more than three terms a polynomial.

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

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials.
Look for the like terms—those with the same variables and the same exponent.
The Commutative Property allows us to rearrange the terms to put like terms together.

Examples

• Find the sum: $(4x^2 + 3x - 7) + (5x^2 - 9x + 3) = (4x^2+5x^2) + (3x-9x) + (-7+3) = 9x^2 - 6x - 4$.
• Find the difference: $(10y^2 - 4y + 2) - (3y^2 + 2y - 6) = 10y^2 - 4y + 2 - 3y^2 - 2y + 6 = 7y^2 - 6y + 8$.
• Subtract $(b^2 - 3b)$ from $(4b^2 + 7b)$: $(4b^2 + 7b) - (b^2 - 3b) = 4b^2 + 7b - b^2 + 3b = 3b^2 + 10b$.

Explanation

To add or subtract polynomials, you just combine like terms on a larger scale. Group the terms with the same variable and exponent, then add or subtract their coefficients. Be extra careful with signs when subtracting!