Lesson 3: Multiply Polynomials

New Concept

Ready to level up your multiplication skills? We'll go beyond basic distribution to multiply various polynomials. You will master multiplying monomials, binomials (using the FOIL method), and trinomials, ensuring every term is accounted for.

What’s next

Get ready to apply these concepts! We'll start with interactive examples of each multiplication type, followed by practice cards to build your confidence and mastery.

Property

We have used the Distributive Property to simplify expressions like $2(x-3)$. You multiplied both terms in the parentheses, $x$ and $3$, by $2$ to get $2x-6$. With this chapter's new vocabulary, you can say you were multiplying a binomial, $x-3$, by a monomial, $2$.

Examples

• To multiply $5(a+6)$, distribute the $5$ to each term: $5 \cdot a + 5 \cdot 6 = 5a + 30$.
• For $3x(2x-5y)$, multiply $3x$ by both terms inside: $3x \cdot 2x - 3x \cdot 5y = 6x^2 - 15xy$.
• To multiply $-4y^2(2y^2 - 3y + 1)$, distribute to all three terms: $-4y^2 \cdot 2y^2 - (-4y^2) \cdot 3y + (-4y^2) \cdot 1 = -8y^4 + 12y^3 - 4y^2$.

Explanation

Think of this as sharing! The monomial outside the parentheses gets 'distributed' or multiplied by every single term inside the polynomial. It's just the Distributive Property you already know, now applied to polynomials.

Property

To multiply $(x+3)(x+7)$, you distribute the second binomial, $(x+7)$, to each term of the first binomial. This gives $x(x+7) + 3(x+7)$. Then, you distribute again to get $x^2 + 7x + 3x + 21$. Finally, combine like terms to get $x^2 + 10x + 21$. Notice that you multiplied the two terms of the first binomial by the two terms of the second binomial, resulting in four multiplications.

Examples

• To multiply $(a+4)(a+6)$, distribute $(a+6)$: $a(a+6) + 4(a+6) = a^2 + 6a + 4a + 24$, which simplifies to $a^2 + 10a + 24$.
• For $(2x+1)(x-3)$, distribute $(x-3)$: $2x(x-3) + 1(x-3) = 2x^2 - 6x + x - 3$, which simplifies to $2x^2 - 5x - 3$.
• To multiply $(y-5)(z+2)$, distribute $(z+2)$: $y(z+2) - 5(z+2) = yz + 2y - 5z - 10$. There are no like terms to combine.

Explanation

This method breaks down the problem into smaller, familiar steps. You take the first term of the first binomial and multiply it by the entire second binomial, then do the same with the second term. It guarantees every piece gets multiplied.

Property

We abbreviate "First, Outer, Inner, Last" as FOIL. The word FOIL is easy to remember and ensures we find all four products.
HOW TO: Multiply a Binomial by a Binomial using the FOIL Method
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.

Examples

• To multiply $(x+2)(x+5)$ with FOIL: First ($x^2$), Outer ($5x$), Inner ($2x$), Last ($10$). The sum is $x^2 + 5x + 2x + 10 = x^2 + 7x + 10$.
• To multiply $(y-3)(2y+1)$ with FOIL: First ($2y^2$), Outer ($y$), Inner ($-6y$), Last ($-3$). The sum is $2y^2 + y - 6y - 3 = 2y^2 - 5y - 3$.
• To multiply $(a-6)(b-2)$ with FOIL: First ($ab$), Outer ($-2a$), Inner ($-6b$), Last ($12$). The result is $ab - 2a - 6b + 12$ since there are no like terms.

Explanation

FOIL is a memory trick for multiplying two binomials. It's a special pattern of the distributive property that ensures you multiply every term in the first binomial by every term in the second one. Just follow the letters!

Property

Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. You line up the binomials vertically, multiply the top binomial by each term of the bottom binomial to get partial products, and then add the partial products, making sure to align like terms in columns.

Examples

• To multiply $(x+5)(x-2)$ vertically, first multiply $-2(x+5)$ to get $-2x-10$. Then multiply $x(x+5)$ to get $x^2+5x$. Add the partial products: $(x^2+5x) + (-2x-10) = x^2+3x-10$.
• To multiply $(3a-4)(a+1)$ vertically, first multiply $1(3a-4)$ to get $3a-4$. Then multiply $a(3a-4)$ to get $3a^2-4a$. Add the partial products: $(3a^2-4a) + (3a-4) = 3a^2-a-4$.
• To multiply $(2y+3)(2y-3)$ vertically, first multiply $-3(2y+3)$ to get $-6y-9$. Then multiply $2y(2y+3)$ to get $4y^2+6y$. Add the partial products: $(4y^2+6y) + (-6y-9) = 4y^2-9$.

Explanation

This is like old-school multiplication with numbers, but for polynomials! You stack the binomials, multiply in parts, and add the results. It's a great way to keep your terms organized, especially with more complex polynomials.

Property

We are now ready to multiply a trinomial by a binomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method. For the Vertical Method, it is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Examples

• Using the Distributive Property: $(x+2)(x^2+3x+1) = x(x^2+3x+1) + 2(x^2+3x+1) = x^3+3x^2+x+2x^2+6x+2 = x^3+5x^2+7x+2$.
• Using the Vertical Method for $(y-3)(y^2-2y+5)$: First, $-3(y^2-2y+5) = -3y^2+6y-15$. Next, $y(y^2-2y+5) = y^3-2y^2+5y$. Adding them gives $y^3 - 5y^2 + 11y - 15$.
• Using the Distributive Property: $(2a-1)(a^2+4a-3) = 2a(a^2+4a-3) - 1(a^2+4a-3) = 2a^3+8a^2-6a-a^2-4a+3 = 2a^3+7a^2-10a+3$.

Explanation

FOIL doesn't work here because there are more than four terms to multiply. Instead, use the Distributive Property or the Vertical Method. Both ensure that every term in the first polynomial multiplies every term in the second.