Engineering A new rectangular outbuilding for a farm is being designed. The outbuilding's side and bottom should be 4 feet thick. Its outer length sho...
Explore Analyzing a Visual Model for Polynomial Factorization
2x 3 + 6x
3
a 3 + b 3 = (a + b)(a 2 - ab + b 2)
7.
Concepts and Skills
x(x + 2)(x + 3) 6.
3
)
x(x(x + 2) + 3(x + 2))
x 3 - 125 = (x - 5)(x 2 + 5x + 25) 8x 3 = (2x)
ASSIGNMENT GUIDE
x (x 2 + 2x) + (3x + 6)
3
a 3 - b 3 = (a - b)(a 2 + ab + b 2)
5.
x 3 + 5x 2 + 6x
x(x 2 + 5x + 6)
3
INTEGRATE TECHNOLOGY Emphasize that students should use caution when checking answers on a graphing calculator. The calculator provides support that the answer is correct, but it cannot be used to prove correctness.
x(x + 2)(x + 8)
Module 6
Lesson 4
316
Exercise
A2_MNLESE385894_U3M06L4.indd 316
Depth of Knowledge (D.O.K.)
Mathematical Practices
1–18
1 Recall of Information
MP.5 Using Tools
19–22
2 Skills/Concepts
MP.4 Modeling
23
1 Recall of Information
MP.2 Reasoning
24
1 Recall of Information
MP.3 Logic
25–26
2 Skills/Concepts
MP.3 Logic
27–29
3 Strategic Thinking
MP.2 Reasoning
3/19/14 1:37 PM
Factoring Polynomials 316
Factor the polynomial by grouping.
AVOID COMMON ERRORS
13. x 3 + 8x 2 + 6x + 48
x (x + 8) + 6(x + 8)
14. x3 + 4x 2 - x - 4
x 2(x + 4) - 1(x + 4)
2
Students may not recognize that a polynomial can sometimes be factored if they regroup the terms. Give students a pattern they can follow to test if factoring by grouping applies to a polynomial: first, rearrange the terms so that when they are grouped, they will have common factors; group the terms; factor each group, using factoring patterns if necessary; then, rearrange and assemble the factors using the distributive property
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 After students have solved a polynomial
19. Engineering A new rectangular outbuilding for a farm is being designed. The outbuilding’s side and bottom should be 4 feet thick. Its outer length should be twice its outer width and height. What should the outer dimensions of the outbuilding be if it is to have a volume of 2304 cubic feet? 2304 = (2x − 8)(x − 8)(x − 4)
The only real solution is x = 16. The outbuilding is 32 feet long, 24 feet wide, and 24 feet high. 20. Arts A piece of rectangular crafting supply is being cut for a new sculpture. You want its length to be 4 times its height and its width to be 2 times its height. If you want the wood to have a volume of 64 cubic centimeters, what will its length, width, and height be?
V = (4x)(2x)(x)
V = 8x 3
64 = 8x 3 8 = x3
2=x
The length of the piece of crafting supply will be 8 cm, the width 4 cm, and the height 2 cm.
Module 6
A2_MNLESE385894_U3M06L4 317
317
Lesson 6.4
(4x 3 - 1)(x - 1)
Write and solve a polynomial equation for the situation described.
equation using the zero-product property, help them understand and recall that the zeros of the polynomial function f(x) associated with the polynomial equation are the values of x where the graph of the polynomial function crosses the x-axis. The zeros of a function f(x) are also equivalent to the solutions of the equation f(x) = 0 and are related to the factors of the polynomial.
