8.5 Factoring Special Products Algebra 1 8.5 Factoring Special Products

Learning Targets Language Goal: Students will be able to describe perfect squares. Math Goal: Students will be able to factor perfect squares and students will be able to factor the difference of two squares. Essential Question: Why are perfect squares helpful when factoring trinomials?

Warm-up

Perfect Square A trinomial is a perfect square if: - the first and last terms are perfect squares - the middle term is two times one factor from the first term and one factor from the last term.   9x² + 12x + 4 3x 3x 2(3x • 2) 2 • 2

𝑥 2 +6𝑥+9= 𝑥+3 𝑥+3 = (𝑥+3) 2 𝑥 2 −2𝑥+1= 𝑥−1 𝑥−1 = (𝑥−1) 2 𝑎 2 +2𝑎𝑏+ 𝑏 2 = 𝑎+𝑏 𝑎+𝑏 = (𝑎+𝑏) 2 𝑎 2 −2𝑎𝑏+ 𝑏 2 = 𝑎−𝑏 𝑎−𝑏 = (𝑎−𝑏) 2

Example Type 1: Recognizing and Factoring Perfect-Square Trinomials Determine whether each trinomial is a perfect square. If so, factor. If not, explain. A. x² + 12x + 36 B. 4x² – 12x + 9

Example Type 1: Recognizing and Factoring Perfect-Square Trinomials Determine whether each trinomial is a perfect square. If so, factor. If not, explain. C. x² + 9x + 16 D. x² + 4x + 4

Example Type 1: Recognizing and Factoring Perfect-Square Trinomials Determine whether each trinomial is a perfect square. If so, factor. If not, explain. E. x² – 14x + 49 F. 9x² – 6x + 4

Example Type 1: Recognizing and Factoring Perfect-Square Trinomials Determine whether each trinomial is a perfect square. If so, factor. If not, explain. G. 9x² – 15x + 64 H. 81x² + 90x + 25 I. 36x² – 10x + 14

Example 2: Word Applications A. The park in the center of the Place des Vosges in Paris, France, is in the shape of a square. The area of the park is (25x² + 70x + 49) ft². The side length of the park is in the form cx + d, where c and d are whole numbers. Find an expression in terms of x for the perimeter of the park. Find the perimeter when x = 8 feet.

Example 2: Word Applications B. A company produces square sheets of aluminum each of which has an area of (9x² + 6x + 1) m². The side length of each sheet is in the form cx + d, where c and d are whole numbers. Find an expression in terms of x for the perimeter of a sheet. Find the perimeter when x = 3 m.

Example 2: Word Applications C. A rectangular piece of cloth must be cut to make a tablecloth. The area needed is (16x² - 24x + 9) ft². The dimensions of the cloth are in the form cx + d, where c and d are whole numbers. Find an expression in terms of x for the perimeter of the cloth. Find the perimeter when x = 11 in.

Difference of Two Squares A polynomial is a difference of two squares if: - There are two terms, one subtracted from the other. - Both terms are perfect squares. 4x² – 9 2x • 2x 3 • 3

Difference of Two Squares 𝑎 2 − 𝑏 2 = 𝑎+𝑏 𝑎−𝑏 𝑥 2 −9= 𝑥+3 𝑥−3

Example Type 3: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. A. x² – 81 B. 9p – 16q²

Example Type 3: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. C. x – 7y² D. 1 – 4x²

Example Type 3: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. E. p – 49q F. 16x² – 4y

Example Type 3: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. G. 3p² – 9q H. 100x² – 4y² I. x – 25y

Closure (𝑥+1) 2 𝑥 2 +2𝑥+1 (𝑥−1) 2 𝑥 2 −2𝑥+1 𝑥+1 (𝑥−1) 𝑥 2 −1

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