How to Multiply Binomials? (+FREE Worksheet!)
Multiply Binomials: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Organize by degreeWrite terms from highest power to lowest power.
- Look for structureTry GCF, special products, grouping, or division depending on the expression.
- Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.
Worked examples
Combine like terms
- Group x squared terms.
- Group x terms.
- Combine each group.
Factor a difference of squares
- Recognize a squared term minus a squared term.
- Use a^2 – b^2.
- Write conjugate factors.
Try one before moving on
Multiply Binomials: pop-up practice
Multiplying binomials is one of the most important skills in Algebra 1 and shows up constantly in factoring, solving equations, and graphing parabolas. The FOIL method — First, Outer, Inner, Last — gives you a reliable shortcut for multiplying any two binomials. This guide covers FOIL step by step, handles special cases like difference of squares and perfect squares, and includes two video lessons with practice problems.
What Is a Binomial?
A binomial is a polynomial with exactly two terms connected by addition or subtraction. Examples: \(\color{blue}{(x + 3)}\), \(\color{blue}{(2x – 5)}\), \(\color{blue}{(a + b)}\). Multiplying two binomials produces a trinomial (three terms) in most cases.
The FOIL Method
FOIL is a mnemonic for the four pairs you multiply when expanding \(\color{blue}{(a + b)(c + d)}\):
- First: multiply the first terms: \(\color{blue}{a \cdot c}\)
- Outer: multiply the outer terms: \(\color{blue}{a \cdot d}\)
- Inner: multiply the inner terms: \(\color{blue}{b \cdot c}\)
- Last: multiply the last terms: \(\color{blue}{b \cdot d}\)
Then combine any like terms (usually the Outer and Inner products).
FOIL Example
\(\color{blue}{(x + 3)(x + 5)}\)
- F: \(\color{blue}{x \cdot x = x^{2}}\)
- O: \(\color{blue}{x \cdot 5 = 5x}\)
- I: \(\color{blue}{3 \cdot x = 3x}\)
- L: \(\color{blue}{3 \cdot 5 = 15}\)
Combine: \(\color{blue}{x^{2} + 5x + 3x + 15 = x^{2} + 8x + 15}\)
Special Products
Difference of Squares
\(\color{blue}{(a + b)(a – b) = a^{2} – b^{2}}\)
- \(\color{blue}{(x + 5)(x – 5) = x^{2} – 25}\)
Perfect Square Trinomial
\(\color{blue}{(a + b)^{2} = a^{2} + 2\text{ ab } + b^{2}}\) and \(\color{blue}{(a – b)^{2} = a^{2}}\) – \(\color{blue}{2\text{ ab } + b^{2}}\)
- \(\color{blue}{(3x – 2)^{2} = 9x^{2} – 12x + 4}\)
Step-by-Step Summary
- Identify the two binomials.
- Apply FOIL: F, O, I, L products.
- Combine the Outer and Inner (like) terms.
- Write the result in descending order.
- Check for special product patterns to save steps.
Watch: Multiplying Binomials (Video Lesson)
Khan Academy explains the distributive approach and FOIL with multiple examples:
Multiplying Binomials — Worked Examples
Example 1: Multiply \(\color{blue}{(x + 3)(x + 5)}\).
F: \(\color{blue}{x^{2}}\) | O: \(\color{blue}{5x}\) | I: \(\color{blue}{3x}\) | L: \(\color{blue}{15}\)
Combine: \(\color{blue}{x^{2} + 8x + 15}\)
Example 2: Multiply \(\color{blue}{(x – 4)(x + 2)}\).
F: \(\color{blue}{x^{2}}\) | O: \(\color{blue}{2x}\) | I: \(\color{blue}{-4x}\) | L: \(\color{blue}{-8}\)
Combine: \(\color{blue}{x^{2} + (2x – 4x) – 8 = x^{2} – 2x – 8}\)
Example 3: Multiply \(\color{blue}{(2x + 3)(3x – 1)}\).
F: \(\color{blue}{6x^{2}}\) | O: \(\color{blue}{-2x}\) | I: \(\color{blue}{9x}\) | L: \(\color{blue}{-3}\)
Combine: \(\color{blue}{6x^{2} + 7x – 3}\)
Example 4: Multiply \(\color{blue}{(x + 5)(x – 5)}\).
Difference of squares pattern: \(\color{blue}{(x + 5)(x – 5) = x^{2} – 5^{2} = x^{2} – 25}\)
Example 5: Expand \(\color{blue}{(3x – 2)^{2}}\).
Perfect square pattern: \(\color{blue}{(3x)^{2} + 2(3x)(-2) + (-2)^{2} = 9x^{2} – 12x + 4}\)
More Practice: Step-by-Step Video
This Khan Academy video demonstrates the binomial multiplication process in detail:
Exercises for Multiplying Binomials
Expand each product and simplify.
- \(\color{blue}{(x + 4)(x + 6)}\)
- \(\color{blue}{(x – 3)(x + 7)}\)
- \(\color{blue}{(2x + 1)(x – 5)}\)
- \(\color{blue}{(3x – 4)(2x + 3)}\)
- \(\color{blue}{(x + 2)^{2}}\)
- \(\color{blue}{(x – 6)(x + 6)}\)
Answers
- \(\color{blue}{x^{2} + 10x + 24}\)
- \(\color{blue}{x^{2} + 4x – 21}\)
- \(\color{blue}{2x^{2} – 9x – 5}\)
- \(\color{blue}{6x^{2} + x – 12}\)
- \(\color{blue}{x^{2} + 4x + 4}\)
- \(\color{blue}{x^{2} – 36}\)
Frequently Asked Questions
What does FOIL stand for?
FOIL stands for First, Outer, Inner, Last — the four pairs of terms you multiply when expanding two binomials. It is a memory device for the full distributive property applied twice.
Does FOIL work for polynomials with more than two terms?
FOIL is designed specifically for two binomials. For longer polynomials, you must distribute each term of the first polynomial to every term of the second, then combine like terms.
What is the difference of squares pattern?
When you multiply conjugate binomials \(\color{blue}{(a + b)(a – b)}\), the middle terms cancel and you get \(\color{blue}{a^{2} – b^{2}}\). Recognizing this saves you from doing FOIL when the pattern fits.
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