Special Products of Binomials 7-8 Special Products of Binomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1

Warm Up Simplify. 1. 42 3. (–2)2 4. (x)2 5. –(5y2) 16 2. 72 49 4 x2 –25y2 6. (m2)2 m4 7. 2(6xy) 12xy 8. 2(8x2) 16x2

Objective Find special products of binomials.

Vocabulary perfect-square trinomial difference of two squares

Imagine a square with sides of length (a + b): The area of this square is (a + b)(a + b) or (a + b)2. The area of this square can also be found by adding the areas of the smaller squares and the rectangles inside. The sum of the areas inside is a2 + ab + ab + b2.

This means that (a + b)2 = a2+ 2ab + b2. You can use the FOIL method to verify this: F L (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 I = a2 + 2ab + b2 O A trinomial of the form a2 + 2ab + b2 is called a perfect-square trinomial. A perfect-square trinomial is a trinomial that is the result of squaring a binomial.

Example 1: Finding Products in the Form (a + b)2 Multiply. A. (x +3)2 Use the rule for (a + b)2. (a + b)2 = a2 + 2ab + b2 Identify a and b: a = x and b = 3. (x + 3)2 = x2 + 2(x)(3) + 32 = x2 + 6x + 9 Simplify. B. (4s + 3t)2

Example 1C: Finding Products in the Form (a + b)2 Multiply. C. (5 + m2)2

Check It Out! Example 1 Multiply. A. (x + 6)2 B. (5a + b)2

Check It Out! Example 1C Multiply. (1 + c3)2

You can use the FOIL method to find products in the form of (a – b)2. (a – b)2 = (a – b)(a – b) = a2 – ab – ab + b2 I O = a2 – 2ab + b2 A trinomial of the form a2 – 2ab + b2 is also a perfect-square trinomial because it is the result of squaring the binomial (a – b).

Example 2: Finding Products in the Form (a – b)2 Multiply. A. (x – 6)2 Use the rule for (a – b)2. (a – b) = a2 – 2ab + b2 Identify a and b: a = x and b = 6. (x – 6) = x2 – 2x(6) + (6)2 = x – 12x + 36 Simplify. B. (4m – 10)2

Example 2: Finding Products in the Form (a – b)2 Multiply. C. (2x – 5y )2 D. (7 – r3)2

Check It Out! Example 2 Multiply. a. (x – 7)2 b. (3b – 2c)2

Check It Out! Example 2c Multiply. (a2 – 4)2

A binomial of the form a2 – b2 is called a difference of two squares. (a + b)(a – b) = a2 – b2 A binomial of the form a2 – b2 is called a difference of two squares.

Example 3: Finding Products in the Form (a + b)(a – b) Multiply. A. (x + 4)(x – 4) Use the rule for (a + b)(a – b). (a + b)(a – b) = a2 – b2 Identify a and b: a = x and b = 4. (x + 4)(x – 4) = x2 – 42 = x2 – 16 Simplify. B. (p2 + 8q)(p2 – 8q)

Example 3: Finding Products in the Form (a + b)(a – b) Multiply. C. (10 + b)(10 – b)

Check It Out! Example 3 Multiply. a. (x + 8)(x – 8) b. (3 + 2y2)(3 – 2y2)

Check It Out! Example 3 Multiply. c. (9 + r)(9 – r)

Example 4: Problem-Solving Application Write a polynomial that represents the area of the yard around the pool shown below.

Check It Out! Example 4 Write an expression that represents the area of the swimming pool.