2.4 Factor and Solve Polynomial Equations p. 111 Name two special factoring patterns for cubes. Name three ways to factor a polynomial. What is the difference between factoring a polynomial and solving a polynomial?

Types of Factoring: From Chapter 1 we did factoring of: –GCF : 6x x = 3x (2x + 5) 1 st on gold card –PTS : x x + 25 = (x + 5) 2 patterns –DOS : 4x 2 – 9 = (2x + 3)(2x – 3) difference of squares –Bustin’ da B = 2x 2 – 5x – 12 = »(2x 2 - 8x) + (3x – 12) = »2x(x – 4) + 3(x – 4)= »(x – 4)(2x + 3)

Factor the polynomial completely. a. x 3 + 2x 2 – 15xFactor common monomial. = x(x + 5)(x – 3) Factor trinomial. b. 2y 5 – 18y 3 Factor common monomial. = 2y 3 (y + 3)(y – 3) Difference of two squares c. 4z 4 – 16z z 2 Factor common monomial. = 4z 2 (z – 2) 2 Perfect square trinomial = x(x 2 + 2x – 15) = 2y 3 (y 2 – 9) = 4z 2 (z 2 – 4z + 4)

Now we will use Sum of Cubes: a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) x = (x) 3 + (2) 3 = (x + 2)(x 2 – 2x + 4)

Difference of Cubes a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) 8x 3 – 1 = (2x) 3 – 1 3 = (2x – 1)((2x) 2 + 2x* ) (2x – 1)(4x 2 + 2x + 1)

When there are more than 3 terms – use GROUPING x 3 – 2x 2 – 9x + 18 = (x 3 – 2x 2 ) + (-9x + 18) = Group in two’s with a ‘+’ in the middle x 2 (x – 2) - 9(x – 2) = GCF each group (x – 2)(x 2 – 9) = (x – 2)(x + 3)(x – 3) Factor all that can be factored

Factor the polynomial completely. a. x = (x + 4)(x 2 – 4x + 16) Sum of two cubes b. 16z 5 – 250z 2 Factor common monomial. = 2z 2 (2z) 3 – 5 3 Difference of two cubes = 2z 2 (2z – 5)(4z z + 25) = x = 2z 2 (8z 3 – 125)

Factor the polynomial completely. 1. x 3 – 7x x SOLUTION x 3 – 7x x= x 3 – 7x x = x( x 2 – 7x + 10) = x( x – 5 )( x – 2 ) Factor common monomial. Factor trinomial.

2. 3y 5 – 75y 3 SOLUTION 3y 5 – 75y 3 = 3y 3 (y 2 – 25) = 3y 3 (y – 5)( y + 5 ) Factor common monomial. Difference of two squares

3. 16b b 2 SOLUTION = 2b 2 (8b ) = 2b 2 (2b + 7)(4b 2 –14b + 49 ) Factor common monomial. Difference of two cubes 4. w 3 – 27 SOLUTION w 3 – 27 = w 3 – (3) 3 = (w – 3)(w 2 + 3w + 9) Difference of two cubes

Factor by Grouping Factor the polynomial x 3 – 3x 2 – 16x + 48 completely. x 3 – 3x 2 – 16x + 48 Factor by grouping. = (x 2 – 16)(x – 3)Distributive property = (x + 4)(x – 4)(x – 3) Difference of two squares = x 2 (x – 3) – 16(x – 3) Can you factor this by using the box method— YES if there are 4 terms

Factoring in Quadratic form: 81x 4 – 16 = (9x 2 ) 2 – 4 2 = (9x 2 + 4)(9x 2 – 4)= Can anything be factored still??? (9x 2 + 4)(3x – 2)(3x +2) Keep factoring ‘till you can’t factor any more!!

You try this one! 4x 6 – 20x x 2 = 4x 2 (x 4 - 5x 2 +6) = 4x 2 (x 2 – 2)(x 2 – 3)

In Chapter 1, we used the zero property. (when multiplying 2 numbers together to get 0 – one must be zero) The also works with higher degree polynomials

Solve: 2x x = 14x 3 2x x x = 0 Put in standard form 2x (x 4 – 7x 2 +12) = 0 GCF 2x (x 2 – 3)(x 2 – 4) = 0 Bustin’ da ‘b’ 2x (x 2 – 3)(x + 2)(x – 2) = 0 Factor everything 2x=0x 2 -3=0x+2=0x-2=0 set all factors to 0 X=0x=±√3x=-2x=2

Now, you try one! 2y 5 – 18y = 0 Y=0y=±√3y=±i√3

City Park You are designing a marble basin that will hold a fountain for a city park. The basin’s sides and bottom should be 1 foot thick. Its outer length should be twice its outer width and outer height. What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?

SOLUTION 36 = (2x – 2)(x – 2)(x – 1) 0 = 2x 3 – 8x x – 40 0 = 2x 2 (x – 4) + 10(x – 4) Write equation. Write in standard form. Factor by grouping. 0 = (2x )(x – 4) Distributive property The only real solution is x = 4. The basin is 8 ft long, 4 ft wide, and 4 ft high. ANSWER

Name two special factoring patterns for cubes. Sum of two cubes and difference of two cubes. Name three ways to factor a polynomial. Factor by sum or difference of cubes, by grouping or when in quadratic form. What is the difference between factoring a polynomial and solving a polynomial? Solving takes the problem one step further than factoring.

2.4 Assignment 2.4 Assignment Page 111, 3 – 48 every third problem