Unit 5 · Lesson 4 of 5
Special products
You'll be able to: square a binomial and multiply \((a + b)(a - b)\) quickly, without the \(x^2 + 9\) mistake.
Step 1 of 5Warm-up
Two quick questions from earlier lessons. Answer from memory first. Pulling it out of your head is what makes it stick.
Multiply out: \((x + 4)(x - 3)\).
FOIL: \(x^2 - 3x + 4x - 12\).
Combine the middle terms: \(-3x + 4x = x\).
Answer: \(x^2 + x - 12\).
Evaluate \(-6^2\).
The exponent goes only on the 6, not on the minus sign: \(-6^2 = -(6 \cdot 6)\).
Answer: \(-36\). (Compare \((-6)^2 = 36\).)
Step 2 of 5The idea
Some products of binomials come up so often that it pays to know the result by sight. You can always get them with FOIL. The patterns just save time.
Square of a sum. \((a + b)^2 = a^2 + 2ab + b^2\)
Square of a difference. \((a - b)^2 = a^2 - 2ab + b^2\)
Sum times difference. \((a + b)(a - b) = a^2 - b^2\)
In words, for a square: first term squared, then twice the product of the two terms, then last term squared. The last term is always \(+\), because a square is never negative.
For \((a + b)(a - b)\), the two middle terms of FOIL are \(+ab\) and \(-ab\). They cancel, so only two terms are left. The answer is a difference of squares.
Why this shows up on the placement test: these patterns are behind factoring, completing the square, and many "which is equivalent" questions later on.
Step 3 of 5Worked examples
Example 1: \((x + 5)^2\), then \((3x - 2)^2\)
- Here \(a = x\) and \(b = 5\). First squared: \(x^2\). Twice the product: \(2 \cdot x \cdot 5 = 10x\). Last squared: \(25\).
- So \((x + 5)^2 = x^2 + 10x + 25\).
- For \((3x - 2)^2\): \(a = 3x\), \(b = 2\). First squared: \((3x)^2 = 9x^2\). Square the 3 too.
- Twice the product: \(2 \cdot 3x \cdot 2 = 12x\), with a minus because it is a difference. Last squared: \(4\).
- So \((3x - 2)^2 = 9x^2 - 12x + 4\).
Example 2: \((2x + 7)(2x - 7)\)
- Same two terms, one plus and one minus, so use \(a^2 - b^2\) with \(a = 2x\), \(b = 7\).
- \((2x)^2 - 7^2 = 4x^2 - 49\). FOIL check: \(4x^2 - 14x + 14x - 49\); the middle terms cancel.
Step 4 of 5Practice
Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went. Use ^ for powers, like x^2 + 6x + 9.
Multiply out: \((x + 6)^2\).
First squared: \(x^2\). Twice the product: \(2 \cdot x \cdot 6 = 12x\). Last squared: \(36\).
Answer: \(x^2 + 12x + 36\).
Multiply out: \((x - 4)^2\).
First squared: \(x^2\). Twice the product: \(2 \cdot x \cdot 4 = 8x\), with a minus. Last squared: \(+16\).
Answer: \(x^2 - 8x + 16\).
Multiply out: \((x + 9)(x - 9)\).
Sum times difference: \(a^2 - b^2\) with \(a = x\), \(b = 9\).
Answer: \(x^2 - 81\).
Multiply out: \((2x + 3)^2\).
First squared: \((2x)^2 = 4x^2\). Twice the product: \(2 \cdot 2x \cdot 3 = 12x\). Last squared: \(9\).
Answer: \(4x^2 + 12x + 9\).
Multiply out: \((3x + 4)(3x - 4)\).
Sum times difference with \(a = 3x\), \(b = 4\): \((3x)^2 - 4^2\).
Answer: \(9x^2 - 16\).
Multiply out: \((5x - 2y)^2\).
\(a = 5x\), \(b = 2y\). First squared: \(25x^2\). Twice the product: \(2 \cdot 5x \cdot 2y = 20xy\), with a minus. Last squared: \(4y^2\).
Answer: \(25x^2 - 20xy + 4y^2\).
Which expression is equal to \((x - 7)^2\)?
- \(x^2 - 14x + 49\)
- \(x^2 - 7x + 49\)
- \(x^2 - 14x - 49\)
- \(x^2 + 14x + 49\)
First squared: \(x^2\). Twice the product: \(2 \cdot x \cdot 7 = 14x\), with a minus. Last squared: \((-7)^2 = +49\).
Answer: \(x^2 - 14x + 49\), choice (a).
(b) forgets the "twice"; (c) gives the last term the wrong sign; (d) gives the middle term the wrong sign.
A square patio has sides \(x\) feet long. Each side is made 3 feet longer. By how many square feet does the area grow? Give a simplified expression.
New area minus old area: \((x + 3)^2 - x^2\).
\((x + 3)^2 = x^2 + 6x + 9\).
\(x^2 + 6x + 9 - x^2 = 6x + 9\).
Answer: \(6x + 9\) square feet.
Step 5 of 5Wrap-up
You can now square a binomial (first squared, twice the product, last squared) and multiply a sum by a difference (\(a^2 - b^2\)), and you know why \((x + 3)^2\) is not \(x^2 + 9\).
Read or watch next: OpenStax Intermediate Algebra 2e, 5.3 Multiply Polynomials (free textbook, with more practice at the end of the section).
Also good: Khan Academy: Algebra 1: Special products of binomials (short videos). Quick rules: Exponents and polynomials cheat sheet.
Something unclear? Ask your teacher bot a follow-up question, for example: "Why is the middle term of \((a + b)^2\) equal to \(2ab\) and not just \(ab\)?" It knows this course and can explain things another way.