Lesson 3: Multiply Polynomials

New Concept

This lesson expands the Distributive Property to multiply various polynomials. You will learn to multiply monomials, binomials using the FOIL method, and trinomials by binomials, mastering different techniques for finding the product of algebraic expressions.

What’s next

Next, you'll see these methods in action with worked examples and short videos. Then, you can test your skills on a series of practice cards.

Property

To multiply a polynomial by a monomial, use the Distributive Property. You multiply each term in the polynomial by the monomial. For an expression like $a(b+c)$, the property states $a(b+c) = ab + ac$.

Examples

• To multiply $5(x+4)$, distribute the $5$ to both $x$ and $4$. This gives $5 \cdot x + 5 \cdot 4$, which simplifies to $5x + 20$.

• For $-3x(2x^2 - 5x + 1)$, distribute $-3x$ to each term inside: $(-3x)(2x^2) + (-3x)(-5x) + (-3x)(1)$, resulting in $-6x^3 + 15x^2 - 3x$.

Property

To multiply binomials, you can use the:

• Distributive Property
• FOIL Method
• Vertical Method

Using the Distributive Property, you multiply each term of the first binomial across the entire second binomial. For $(a+b)(c+d)$, you distribute $(c+d)$ to get $a(c+d) + b(c+d)$, then distribute again to get $ac + ad + bc + bd$.

Examples

• Using the Distributive Property for $(x+2)(x+5)$, we distribute $(x+5)$ to get $x(x+5) + 2(x+5)$. Distributing again gives $x^2 + 5x + 2x + 10$, which combines to $x^2 + 7x + 10$.

Property

Use the FOIL method for multiplying two binomials.
Step 1. Multiply the First terms.
Step 2. Multiply the Outer terms.
Step 3. Multiply the Inner terms.
Step 4. Multiply the Last terms.
Step 5. Combine like terms, when possible.
For $(a+b)(c+d)$, the products are: First ($ac$), Outer ($ad$), Inner ($bc$), Last ($bd$).

Examples

• To multiply $(x+3)(x+8)$ with FOIL: First is $x \cdot x = x^2$. Outer is $x \cdot 8 = 8x$. Inner is $3 \cdot x = 3x$. Last is $3 \cdot 8 = 24$. Combine to get $x^2 + 8x + 3x + 24 = x^2 + 11x + 24$.

• For $(y-7)(y+2)$: F gives $y^2$, O gives $2y$, I gives $-7y$, and L gives $-14$. The expression is $y^2 + 2y - 7y - 14$. Combining like terms simplifies this to $y^2 - 5y - 14$.

Property

The Vertical Method for multiplying polynomials is analogous to multiplying whole numbers. Write one polynomial above the other. Multiply the top polynomial by each term of the bottom polynomial, creating partial products. Align like terms in columns and add the partial products to get the final answer.

Examples

• To multiply $(2x-1)(3x+5)$ vertically, write $2x-1$ above $3x+5$. First, $5(2x-1) = 10x-5$. Below that, write $3x(2x-1) = 6x^2-3x$, aligning terms. Adding the columns gives $6x^2+7x-5$.

• Let's multiply $(4y-3)(y-2)$. Write $4y-3$ on top. The first partial product is $-2(4y-3) = -8y+6$. The second is $y(4y-3) = 4y^2-3y$. Adding them gives $4y^2 - 11y + 6$.

Property

To multiply a trinomial by a binomial, the FOIL method will not work. You must use either the Distributive Property or the Vertical Method. Using the Distributive Property, you distribute each term of the binomial to the entire trinomial. For $(a+b)(c+d+e)$, this becomes $a(c+d+e) + b(c+d+e)$.

Examples

• Using the Distributive Property for $(x+2)(x^2+4x-5)$: $x(x^2+4x-5) + 2(x^2+4x-5)$. This expands to $x^3+4x^2-5x + 2x^2+8x-10$. Combining like terms gives $x^3+6x^2+3x-10$.

• Using the Vertical Method for $(y-3)(y^2-5y+8)$: Multiply $y^2-5y+8$ by $-3$ to get $-3y^2+15y-24$. Then multiply by $y$ to get $y^3-5y^2+8y$. Adding the aligned terms gives $y^3-8y^2+23y-24$.