Graph the system of linear inequalities. y > –x + 5

Objective Find special products of binomials.

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.

= x2 + 6x + 9 = 16s2 + 24st + 9t2 Multiply. A. (x +3)2 (a + b)2 = a2 + 2ab + b2 = x2 + 6x + 9 B. (4s + 3t)2 = 16s2 + 24st + 9t2

Multiply. C. (5 + m2)2 (a + b)2 = a2 + 2ab + b2 = 25 + 10m2 + m4

= x2 + 12x + 36 = 25a2 + 10ab + b2 Multiply. A. (x + 6)2 (a + b)2 = a2 + 2ab + b2 = x2 + 12x + 36 B. (5a + b)2 = 25a2 + 10ab + b2

Multiply. (1 + c3)2 (a + b)2 = a2 + 2ab + b2 = 1 + 2c3 + c6

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).

= x – 12x + 36 = 16m2 – 80m + 100 Multiply. (a – b)2 = a2 – 2ab + b2 A. (x – 6)2 = x – 12x + 36 B. (4m – 10)2 = 16m2 – 80m + 100

= 4x2 – 20xy +25y2 = 49 – 14r3 + r6 Multiply. C. (2x – 5y )2 (a – b) = a2 – 2ab + b2 = 4x2 – 20xy +25y2 D. (7 – r3)2 = 49 – 14r3 + r6

You can use an area model to see that (a + b)(a - b) = a2 – b2. Begin with a square with area a2. Remove a square with area b2. The area of the new figure is a2 – b2. Then remove the smaller rectangle on the bottom. Turn it and slide it up next to the top rectangle. The new arrange- ment is a rectangle with length a + b and width a – b. Its area is (a + b)(a – b). So (a + b)(a – b) = a2 – b2. A binomial of the form a2 – b2 is called a difference of two squares.

= x2 – 16 = p4 – 64q2 Multiply. A. (x + 4)(x – 4) (a + b)(a – b) = a2 – b2 = x2 – 16 B. (p2 + 8q)(p2 – 8q) = p4 – 64q2

Multiply. C. (10 + b)(10 – b) (a + b)(a – b) = a2 – b2 = 100 – b2

Multiply. a. (x + 8)(x – 8) = x2 – 64 b. (3 + 2y2)(3 – 2y2) = 9 – 4y4

Multiply. c. (9 + r)(9 – r) = 81 – r2

= x2 + 10x + 25 = x2 – 4 x2 + 10x + 25 – (x2 – 4) = 10x + 29 Write a polynomial that represents the area of the yard around the pool shown below. (a + b)2 = a2 + 2ab + b2 (a + b)(a – b) = a2 – b2 = x2 + 10x + 25 = x2 – 4 x2 + 10x + 25 – (x2 – 4) = 10x + 29

Write an expression that represents the area of the swimming pool. 25 – x2 + x2 25