Product Of Two Binomials Examples With Answers

The product of two binomials examples with answers shows how to multiply expressions such as ((x+5)(x‑4)) and obtain a simplified quadratic result, making this fundamental algebra skill accessible to learners of all levels.

Introduction

Understanding the product of two binomials is a cornerstone of algebra because it bridges simple arithmetic and the more complex world of quadratic equations. When you multiply two binomials, you are essentially applying the distributive property twice, which can be streamlined using the FOIL technique (First, Outer, Inner, Last). This method not only speeds up calculations but also helps students recognize patterns that appear in factoring, solving equations, and even in real‑world applications like physics and economics. In this article we will break down the process step by step, provide clear examples with answers, and address common questions that arise during learning.

Steps to Multiply Two Binomials

Identify the terms

1. Write each binomial in standard form (e.g., (a + b) and (c + d)).
2. Label the four products you will create:
   • First: multiply the first terms of each binomial.
   • Outer: multiply the outer terms.
   • Inner: multiply the inner terms.
   • Last: multiply the last terms of each binomial.

Apply the FOIL method

• First terms → (a \times c)
• Outer terms → (a \times d)
• Inner terms → (b \times c)
• Last terms → (b \times d)

Combine like terms

Add the four products together and simplify by merging any like terms (terms with the same variable raised to the same power) It's one of those things that adds up. That's the whole idea..

Verify the result

Optionally, expand the result by distributing again to ensure no mistakes were made.

Example 1

Multiply ((x + 3)(x - 2)).

• First: (x \times x = x^{2})
• Outer: (x \times (-2) = -2x)
• Inner: (3 \times x = 3x)
• Last: (3 \times (-2) = -6

… = (-6)

Now combine the like terms (-2x) and (3x):

[ x^{2} - 2x + 3x - 6 = x^{2} + x - 6 ]

Answer: ((x + 3)(x - 2) = x^{2} + x - 6)

Example 2

Multiply ((2x + 1)(3x - 4)) That's the part that actually makes a difference..

• First: (2x \times 3x = 6x^{2})
• Outer: (2x \times (-4) = -8x)
• Inner: (1 \times 3x = 3x)
• Last: (1 \times (-4) = -4)

Combine like terms:

[ 6x^{2} - 8x + 3x - 4 = 6x^{2} - 5x - 4 ]

Answer: ((2x + 1)(3x - 4) = 6x^{2} - 5x - 4)

Example 3

Multiply ((x + 5)(x - 5)).

• First: (x \times x = x^{2})
• Outer: (x \times (-5) = -5x)
• Inner: (5 \times x = 5x)
• Last: (5 \times (-5) = -25)

Combine like terms:

[ x^{2} - 5x + 5x - 25 = x^{2} - 25 ]

Notice the middle terms cancel out. This is a difference of squares And it works..

Answer: ((x + 5)(x - 5) = x^{2} - 25)

Example 4

Multiply ((x - 7)^{2}), which is the same as ((x - 7)(x - 7)).

• First: (x \times x = x^{2})
• Outer: (x \times (-7) = -7x)
• Inner: ((-7) \times x = -7x)
• Last: ((-7) \times (-7) = 49)

Combine like terms:

[ x^{2} - 7x -