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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
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
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
𝑥2+2𝑥+1
𝑥2−2 𝑥+1
𝑥2−1
(𝑥+1)2
(𝑥−1)2
(𝑥+1 )(𝑥−1)
Lesson Quiz