Factoring Special Products

Slides:



Advertisements
Similar presentations
7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
Advertisements

5 Minute Warm-Up Directions: Simplify each problem. Write the
Objectives Factor the difference of two squares..
Factoring Special Products
Warm Up Determine whether the following are perfect squares. If so, find the square root. 64 yes; yes; no 4. x2 yes; x 5. y8 yes; y4 6.
Factoring Polynomials
 Polynomials Lesson 5 Factoring Special Polynomials.
Special Products of Binomials
Objectives Choose an appropriate method for factoring a polynomial.
Warm Up Factor each expression. 1. 3x – 6y 3(x – 2y) 2. a2 – b2
Special Products of Binomials
Factoring Polynomials
Objectives Use the Factor Theorem to determine factors of a polynomial. Factor the sum and difference of two cubes.
Objectives Use the Factor Theorem to determine factors of a polynomial. Factor the sum and difference of two cubes.
Holt McDougal Algebra Multiplying Polynomials 7-8 Multiplying Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation.
Solving Quadratics: Factoring. What is a factor? Numbers you can multiply to get another number 2  3.
Preview Warm Up California Standards Lesson Presentation.
Algebra I Review of Factoring Polynomials
Objectives Factor perfect-square trinomials.
8.5 Factoring Special Products
Special Products of Binomials
Special Products of Binomials
Warm Up Determine whether the following are perfect squares. If so, find the square root. 64 yes; yes; no 4. x2 yes; x 5. y8 yes; y4 6.
Choosing a Factoring Method
Objectives Factor the sum and difference of two cubes.
Holt McDougal Algebra Special Products of Binomials Warm Up Simplify (–2) 2 4. (x) 2 5. –(5y 2 ) x2x (m 2 ) 2 m4m4.
ALGEBRA 1 Lesson 8-7 Warm-Up ALGEBRA 1 “Factoring Special Cases” (8-7) What is a “perfect square trinomial”? How do you factor a “perfect square trinomial”?
Factoring Polynomials.
7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
Warm Up 01/18/17 Simplify (–2)2 3. –(5y2) 4. 2(6xy)
Objectives Factor perfect-square trinomials.
Factoring Polynomials
Welcome! Grab a set of interactive notes and study Guide
Objectives Factor perfect-square trinomials.
Choosing a Factoring Method
Warm Up 1.) What is the factored form of 2x2 + 9x + 10? 2.) What is the factored form of 6x2 – 23x – 4?
Determine whether the following are perfect squares
Factoring Special Products
Do Now Determine if the following are perfect squares. If yes, identify the positive square root /16.
6-3 Polynomials Warm Up Lesson Presentation Lesson Quiz
Factoring Polynomials
Factoring Special Products
Factoring Polynomials
Factoring Polynomials
8-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
Lesson 9.7 Factor Special Products
Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
Choosing a Factoring Method
Choosing a Factoring Method
Special Products of Binomials
Warm Up: Solve the Equation 4x2 + 8x – 32 = 0.
Factoring Polynomials
7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz
Factoring the Difference of
Lesson 7-5 Factoring Special Products Lesson 7-6 Choosing a Factoring Method Obj: The student will be able to 1) Factor perfect square trinomials 2) Factor.
8-8 Completing the Square Warm Up Lesson Presentation Lesson Quiz
Choosing a Factoring Method
Objectives Factor perfect-square trinomials.
College Algebra.
Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
Factoring Special Products
Warm Up Factor the following: a) b).
Choosing a Factoring Method
Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz
Choosing a Factoring Method
Choosing a Factoring Method
Presentation transcript:

Factoring Special Products 7-5 Factoring Special Products Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Objectives Factor perfect-square trinomials. Factor the difference of two squares.

Example 1B: Recognizing and Factoring Perfect-Square Trinomials Determine whether each trinomial is a perfect square. If so, factor. If not explain. 81x2 + 90x + 25 81x2 + 90x + 25 5 5 9x 9x 2(9x 5) ● The trinomial is a perfect square. Factor.

Example 1A: Recognizing and Factoring Perfect-Square Trinomials Determine whether each trinomial is a perfect square. If so, factor. If not explain. 9x2 – 15x + 64 9x2 – 15x + 64 8 8 3x 3x 2(3x 8)  2(3x 8) ≠ –15x.  9x2 – 15x + 64 is not a perfect-square trinomial because –15x ≠ 2(3x  8).

Example 1B Continued Determine whether each trinomial is a perfect square. If so, factor. If not explain. Method 2 Use the rule. 81x2 + 90x + 25 a = 9x, b = 5 (9x)2 + 2(9x)(5) + 52 Write the trinomial as a2 + 2ab + b2. Write the trinomial as (a + b)2. (9x + 5)2

The trinomial is a perfect square. Factor. Check It Out! Example 1a Determine whether each trinomial is a perfect square. If so, factor. If not explain. x2 + 4x + 4 x x 2 2  2(x 2) x2 + 4x + 4 The trinomial is a perfect square. Factor.

Check It Out! Example 1a Continued Determine whether each trinomial is a perfect square. If so, factor. If not explain. Method 1 Factor. x2 + 4x + 4 Factors of 4 Sum (1 and 4) 5 (2 and 2) 4   (x + 2)(x + 2) = (x + 2)2

The trinomial is a perfect square. Factor. Check It Out! Example 1b Determine whether each trinomial is a perfect square. If so, factor. If not explain. x2 – 14x + 49 x2 – 14x + 49 x x 2(x 7) 7 7  The trinomial is a perfect square. Factor.

Check It Out! Example 1b Continued Determine whether each trinomial is a perfect square. If so, factor. If not explain. Method 2 Use the rule. x2 – 14x + 49 a = 1, b = 7 (x)2 – 2(x)(7) + 72 Write the trinomial as a2 – 2ab + b2. (x – 7)2 Write the trinomial as (a – b)2.

Check It Out! Example 2 What if …? 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.

Understand the Problem Check It Out! Example 2 Continued 1 Understand the Problem The answer will be an expression for the perimeter of a sheet and the value of the expression when x = 3. List the important information: A sheet is a square with area (9x2 + 6x + 1) m2. The side length of a sheet is in the form cx + d, where c and d are whole numbers.

Check It Out! Example 2 Continued Make a Plan The formula for the area of a sheet is area = (side)2 Factor 9x2 + 6x + 1 to find the side length of a sheet. Write a formula for the perimeter of the sheet, and evaluate the expression for x = 3.

Check It Out! Example 2 Continued Solve 3 9x2 + 6x + 1 a = 3x, b = 1 (3x)2 + 2(3x)(1) + 12 Write the trinomial as a2 + 2ab + b2. (3x + 1)2 Write the trinomial as (a + b)2. 9x2 + 6x + 1 = (3x + 1)(3x + 1) The side length of a sheet is (3x + 1) m.

Check It Out! Example 2 Continued Write a formula for the perimeter of the aluminum sheet. Write the formula for the perimeter of a square. P = 4s = 4(3x + 1) Substitute the side length for s. = 12x + 4 Distribute 4. An expression for the perimeter of the sheet in meters is 12x + 4.

Check It Out! Example 2 Continued Evaluate the expression when x = 3. P = 12x + 4 = 12(3) + 4 Substitute 3 for x. = 40 When x = 3 m. the perimeter of the sheet is 40 m.

Check It Out! Example 2 Continued 4 Look Back For a square with a perimeter of 40, the side length is m and the area is 102 = 100 m2. Evaluate 9x2 + 6x + 1 for x = 3 9(3)2 + 6(3) + 1 81 + 18 + 1 100 

Example 3B: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 100x2 – 4y2 100x2 – 4y2 2y 2y  10x 10x The polynomial is a difference of two squares. (10x)2 – (2y)2 a = 10x, b = 2y (10x + 2y)(10x – 2y) Write the polynomial as (a + b)(a – b). 100x2 – 4y2 = (10x + 2y)(10x – 2y)

Example 3C: Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. x4 – 25y6 x4 – 25y6 5y3 5y3  x2 x2 The polynomial is a difference of two squares. (x2)2 – (5y3)2 a = x2, b = 5y3 Write the polynomial as (a + b)(a – b). (x2 + 5y3)(x2 – 5y3) x4 – 25y6 = (x2 + 5y3)(x2 – 5y3)

The polynomial is a difference of two squares. Check It Out! Example 3a Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 1 – 4x2 1 – 4x2 2x 2x  1 1 The polynomial is a difference of two squares. (1) – (2x)2 a = 1, b = 2x (1 + 2x)(1 – 2x) Write the polynomial as (a + b)(a – b). 1 – 4x2 = (1 + 2x)(1 – 2x)

The polynomial is a difference of two squares. Check It Out! Example 3b Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. p8 – 49q6 p8 – 49q6 7q3 7q3 – p4 p4 The polynomial is a difference of two squares. (p4)2 – (7q3)2 a = p4, b = 7q3 (p4 + 7q3)(p4 – 7q3) Write the polynomial as (a + b)(a – b). p8 – 49q6 = (p4 + 7q3)(p4 – 7q3)

Lesson Quiz: Part I Determine whether each trinomial is a perfect square. If so factor. If not, explain. 64x2 – 40x + 25 2. 121x2 – 44x + 4 3. 49x2 + 140x + 100 4. A fence will be built around a garden with an area of (49x2 + 56x + 16) ft2. The dimensions of the garden are cx + d, where c and d are whole numbers. Find an expression for the perimeter when x = 5. Not a perfect-square trinomial because –40x ≠ 2(8x  5). (11x – 2)2 (7x2 + 10)2 P = 28x + 16; 156 ft

Lesson Quiz: Part II Determine whether the binomial is a difference of two squares. If so, factor. If not, explain. 5. 9x2 – 144y4 6. 30x2 – 64y2 7. 121x2 – 4y8 (3x + 12y2)(3x – 12y2) Not a difference of two squares; 30x2 is not a perfect square (11x + 2y4)(11x – 2y4)