Learn on PengiCalifornia Reveal Math, Algebra 1Unit 9: Polynomials

9-4 Special Products

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn to multiply binomials using three special product patterns: the Square of a Sum, the Square of a Difference, and the Product of a Sum and a Difference. By recognizing these patterns, students can simplify expressions like (a + b)², (a − b)², and (a + b)(a − b) = a² − b² more efficiently than applying the Distributive Property each time. The lesson is part of Unit 9: Polynomials and builds on students' prior knowledge of multiplying binomials.

Section 1

Square of a sum

The square of a binomial sum follows the pattern: (a+b)2=(a+b)(a+b)=a2+2ab+b2(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2.

Example 1: (4p+3q)2=(4p)2+2(4p)(3q)+(3q)2=16p2+24pq+9q2(4p + 3q)^2 = (4p)^2 + 2(4p)(3q) + (3q)^2 = 16p^2 + 24pq + 9q^2.
Example 2: (x+5)2=x2+2(x)(5)+52=x2+10x+25(x + 5)^2 = x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25.

Squaring a binomial isn't just squaring each piece inside! You're multiplying the binomial by itself, which requires the FOIL method. This pattern, a2+2ab+b2a^2 + 2ab + b^2, is a fantastic time-saver. It reminds you to always include the product of the 'Outside' and 'Inside' terms, which are always identical. This middle term is the secret ingredient!

Section 2

Square of a difference

The square of a binomial difference follows the pattern: (a−b)2=(a−b)(a−b)=a2−2ab+b2(a - b)^2 = (a - b)(a - b) = a^2 - 2ab + b^2.

Example 1: (3y−7)2=(3y)2−2(3y)(7)+72=9y2−42y+49(3y - 7)^2 = (3y)^2 - 2(3y)(7) + 7^2 = 9y^2 - 42y + 49.
Example 2: (k−6)2=k2−2(k)(6)+62=k2−12k+36(k - 6)^2 = k^2 - 2(k)(6) + 6^2 = k^2 - 12k + 36.

Just like squaring a sum, squaring a difference has a special pattern to save you time. When you multiply (a−b)(a-b) by itself, the 'Outside' and 'Inside' terms are both negative, combining to create the middle term −2ab-2ab. Remembering this formula helps you avoid common errors and solve these problems in a single, confident step without writing out FOIL.

Section 3

Sum and difference

The product of the sum and difference of the same two terms follows the pattern: (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2.

Example 1: (3x+4y)(3x−4y)=(3x)2−(4y)2=9x2−16y2(3x + 4y)(3x - 4y) = (3x)^2 - (4y)^2 = 9x^2 - 16y^2.
Example 2: (z+8)(z−8)=z2−82=z2−64(z + 8)(z - 8) = z^2 - 8^2 = z^2 - 64.

This is the ultimate shortcut for multiplying conjugate pairs! When you see two binomials with the exact same terms but opposite signs, the 'Outside' and 'Inside' products magically cancel each other out. This pattern lets you jump straight to the answer: just square the first term, square the second term, and put a minus sign between them!

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Square of a sum

The square of a binomial sum follows the pattern: (a+b)2=(a+b)(a+b)=a2+2ab+b2(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2.

Example 1: (4p+3q)2=(4p)2+2(4p)(3q)+(3q)2=16p2+24pq+9q2(4p + 3q)^2 = (4p)^2 + 2(4p)(3q) + (3q)^2 = 16p^2 + 24pq + 9q^2.
Example 2: (x+5)2=x2+2(x)(5)+52=x2+10x+25(x + 5)^2 = x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25.

Squaring a binomial isn't just squaring each piece inside! You're multiplying the binomial by itself, which requires the FOIL method. This pattern, a2+2ab+b2a^2 + 2ab + b^2, is a fantastic time-saver. It reminds you to always include the product of the 'Outside' and 'Inside' terms, which are always identical. This middle term is the secret ingredient!

Section 2

Square of a difference

The square of a binomial difference follows the pattern: (a−b)2=(a−b)(a−b)=a2−2ab+b2(a - b)^2 = (a - b)(a - b) = a^2 - 2ab + b^2.

Example 1: (3y−7)2=(3y)2−2(3y)(7)+72=9y2−42y+49(3y - 7)^2 = (3y)^2 - 2(3y)(7) + 7^2 = 9y^2 - 42y + 49.
Example 2: (k−6)2=k2−2(k)(6)+62=k2−12k+36(k - 6)^2 = k^2 - 2(k)(6) + 6^2 = k^2 - 12k + 36.

Just like squaring a sum, squaring a difference has a special pattern to save you time. When you multiply (a−b)(a-b) by itself, the 'Outside' and 'Inside' terms are both negative, combining to create the middle term −2ab-2ab. Remembering this formula helps you avoid common errors and solve these problems in a single, confident step without writing out FOIL.

Section 3

Sum and difference

The product of the sum and difference of the same two terms follows the pattern: (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2.

Example 1: (3x+4y)(3x−4y)=(3x)2−(4y)2=9x2−16y2(3x + 4y)(3x - 4y) = (3x)^2 - (4y)^2 = 9x^2 - 16y^2.
Example 2: (z+8)(z−8)=z2−82=z2−64(z + 8)(z - 8) = z^2 - 8^2 = z^2 - 64.

This is the ultimate shortcut for multiplying conjugate pairs! When you see two binomials with the exact same terms but opposite signs, the 'Outside' and 'Inside' products magically cancel each other out. This pattern lets you jump straight to the answer: just square the first term, square the second term, and put a minus sign between them!