Lesson 1: Add and Subtract Polynomials

New Concept

This lesson introduces polynomials, the building blocks of algebra. You'll learn to identify their types and degrees, then master the core skills of adding, subtracting, and evaluating polynomial expressions for given values.

What’s next

Now you have the basics. Up next, you'll apply these concepts through a series of practice cards, interactive examples, and challenge problems.

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. A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.

Examples

• The degree of $8x^5 - 3x^2 + 1$ is $5$, because the term with the highest exponent is $8x^5$.
• The degree of $4x^3y^2 + 7xy$ is $5$, because the sum of exponents in the first term ($3+2=5$) is the highest.
• The degree of the constant $-20$ is $0$.

Explanation

The degree tells you the 'power' of a polynomial. Find the degree of each term, and the largest one becomes the degree for the whole polynomial. Writing terms from the highest degree to the lowest is the standard form.

Property

Adding and subtracting monomials is the same as combining like terms. Remember, like terms must have the same variables with the same exponents. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

Examples

• To add $12x^2$ and $8x^2$, we combine the like terms: $12x^2 + 8x^2 = 20x^2$.
• To subtract $4y$ from $15y$, we perform the operation: $15y - 4y = 11y$.
• The monomials $5a^2$ and $5b^2$ cannot be combined because their variables are different.

Explanation

You can only combine monomials that are 'alike'—meaning they have the same variable and exponent. Just add or subtract the numbers in front (coefficients) and keep the variable part identical. It is like adding apples to apples.

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!

Property

To evaluate a polynomial, we substitute the given value for the variable and then simplify using the order of operations.

Examples

• Evaluate $4x^2 - 3x + 2$ for $x=3$. Substitute $x=3$: $4(3)^2 - 3(3) + 2 = 4(9) - 9 + 2 = 36 - 9 + 2 = 29$.
• Evaluate $2y^3 + 5y$ for $y=-2$. Substitute $y=-2$: $2(-2)^3 + 5(-2) = 2(-8) - 10 = -16 - 10 = -26$.
• The cost of producing a widget is given by $5x^2 + 10x$. For $x=5$ widgets, the cost is $5(5)^2 + 10(5) = 5(25) + 50 = 125 + 50 = 175$ dollars.

Explanation

To evaluate a polynomial, simply replace the variable with the given number. After substituting, follow the order of operations (PEMDAS) to calculate the final numerical answer. It turns an expression into a single value.