WarmUp Exercises Factor completely 1 x 2 x

Warm-Up Exercises Factor completely. 1. x 2 – x – 12 ANSWER (x – 4)(x + 3) 2. 2 x 2 – 5 x – 3 ANSWER (x – 3)(2 x + 1)

Warm-Up Exercises Factor completely. 3. Use synthetic division to divide 2 x 3 – 3 x 2 – 18 x – 8 by x – 4. ANSWER 2 x 2 + 5 x + 2 4. The volume of a box is modeled by f (x) = x (x – 1)(x – 2), where x is the length in meters. What is the volume when the length is 3 meters. ANSWER 6 m 3

Warm-Up 1 Exercises EXAMPLE List possible rational zeros List the possible rational zeros of f using the rational zero theorem. a. f (x) = x 3 + 2 x 2 – 11 x + 12 Factors of the constant term: + 1, + 2, + 3, + 4, + 6, + 12 Factors of the leading coefficient: + 1 Possible rational zeros: + 1 , + 2 , + 3 , + 4 , + 6 , + 12 1 1 1 Simplified list of possible zeros: + 1, + 2, + 3, + 4, + 6, + 12

List possible rational zeros Warm-Up 1 Exercises EXAMPLE b. f (x) = 4 x 4 – x 3 – 3 x 2 + 9 x – 10 Factors of the constant term: + 1, + 2, + 5, + 10 Factors of the leading coefficient: + 1, + 2, + 4 Possible rational zeros: + 1 , + 2 , + 5 , + 10 , + 1 , + 2 , + 5 4 2 2 4 1 1 + 10 4 Simplified list of possible zeros: + 1, + 2, + 5, + 10, + 1 2 + 5 , + 1 , + 5 2 4 4

Warm-Up Exercises GUIDED PRACTICE for Example 1 List the possible rational zeros of f using the rational zero theorem. 1. f (x) = x 3 + 9 x 2 + 23 x + 15 SOLUTION Factors of the constant term: + 1, + 3, + 5, + 15 Factors of the leading coefficient: + 1 Possible rational zeros: + 1 , + 3 , + 5 , + 15 1 1 Simplified list of possible zeros: + 1, + 3, + 5 + 15

Warm-Up Exercises GUIDED PRACTICE for Example 1 2. f (x) =2 x 3 + 3 x 2 – 11 x – 6 SOLUTION Factors of the constant term: + 1, + 2, + 3 Factors of the leading coefficient: + 1, + 2, + 4 Possible rational zeros: + 1 , + 2 , + 3 , +6 , + 1 , + 3 2 2 1 1 Simplified list of possible zeros: + 1, + 2, + 3, + 6 + 1 2 + 3 2

Find zeros when the leading coefficient is 1 Warm-Up 2 Exercises EXAMPLE Find all real zeros of f (x) = x 3 – 8 x 2 +11 x + 20. SOLUTION STEP 1 List the possible rational zeros. The leading coefficient is 1 and the constant term is 20. So, the possible rational zeros are: x = + 1 , + 2 , + 4 , + 5 , + 10 , + 20 1 1 1

Find zeros when the leading coefficient is 1 Warm-Up 2 Exercises EXAMPLE STEP 2 Test these zeros using synthetic division. Test x =1: 1 1 – 8 11 1 – 7 4 1 Test x = – 1: – 1 1 – 8 1 – 9 11 20 4 24 ↑ 1 is not a zero. 20 9 20 20 0 ↑ – 1 is a zero

Find zeros when the leading coefficient is 1 Warm-Up 2 Exercises EXAMPLE Because – 1 is a zero of f, you can write f (x) = (x + 1)(x 2 – 9 x + 20). STEP 3 Factor the trinomial in f (x) and use the factor theorem. f (x) = (x + 1) (x 2 – 9 x + 20) = (x + 1)(x – 4)(x – 5) ANSWER The zeros of f are – 1, 4, and 5.

Warm-Up Exercises GUIDED PRACTICE for Example 2 Find all real zeros of the function. 3. f (x) = x 3 – 4 x 2 – 15 x + 18 SOLUTION Factors of the constant term: + 1, + 2, + 3, + 6, + 9 Factors of the leading coefficient: + 1 Possible rational zeros: + 1 , + 3 + 6 1 1 1 Simplified list of possible zeros: – 3, 1 , 6

Warm-Up Exercises GUIDED PRACTICE 4. for Example 2 f (x) + x 3 – 8 x 2 + 5 x+ 14 SOLUTION STEP 1 List the possible rational zeros. The leading coefficient is 1 and the constant term is 14. So, the possible rational zeros are: x=+ 1 , + 2, + 7 1 1 1

Warm-Up Exercises GUIDED PRACTICE STEP 2 for Example 2 Test these zeros using synthetic division. Test x =1: 1 1 – 8 1 – 7 1 5 14 – 7 – 2 12 ↑ Test x = – 1: – 1 1 1 – 8 5 14 – 1 – 9 9 14 14 0 ↑ 1 is not a zero. – 1 is a zero.

Warm-Up Exercises GUIDED PRACTICE for Example 2 Because – 1 is a zero of f, you can write f (x) = (x + 1)(x 2 – 9 x + 14). STEP 3 Factor the trinomial in f (x) and use the factor theorem. f (x) = (x + 1) (x 2 – 9 x + 14) = (x + 1)(x + 4)(x – 7) ANSWER The zeros of f are – 1, 2, and 7.

Find zeros when the leading coefficient is not 1 Warm-Up 3 Exercises EXAMPLE Find all real zeros of f (x) =10 x 4 – 11 x 3 – 42 x 2 + 7 x + 12. SOLUTION STEP 1 List the possible rational zeros of f : + 1 , + 2 , + 3 , + 4 , + 6 , + 12 , + 1 , + 3 , + 1 1 1 2 5 , + 2 , + 3 , + 4 , + 6 , + 12 , + 1 , + 3 5 5 5 10 10

Find zeros when the leading coefficient is not 1 Warm-Up 3 Exercises EXAMPLE STEP 2 Choose reasonable values from the list above to check using the graph of the function. For f , the values x = – 3 , x = 1 , x = 3 , and x = 12 2 5 are reasonable based on the graph shown at the right.

Find zeros when the leading coefficient is not 1 Warm-Up 3 Exercises EXAMPLE STEP 3 Check the values using synthetic division until a zero is found. – 3 10 – 11 – 42 7 12 2 9 – 69 – 15 39 4 2 21 10 – 26 – 3 23 – 4 2 10 – 11 – 42 7 12 10 – 5 – 16 8 17 – 12 – 34 24 0 – ↑ 1 2 is a zero.

Find zeros when the leading coefficient is not 1 Warm-Up 3 Exercises EXAMPLE STEP 4 Factor out a binomial using the result of the synthetic division. f (x) = x + 1 2 (10 x 3 – 16 x 2 – 34 x + 24) Write as a product of factors. = x + 1 (2)(5 x 3 – 8 x 2 – 17 x + 12) Factor 2 out of the 2 second factor. = (2 x +1)(5 x 3 – 8 x 2 – 17 x +12) Multiply the first factor by 2.

Find zeros when the leading coefficient is not 1 Warm-Up 3 Exercises EXAMPLE STEP 5 Repeat the steps above for g (x) = 5 x 3 – 8 x 2 – 17 x + 12. Any zero of g will also be a zero of f. The possible rational zeros of g are: x = + 1, + 2, + 3, + 4, + 6, + 12, + 1 , + 2 , + 3 , + 4 , + 6 , + 12 5 5 5 3 The graph of g shows that 5 may be a zero. 3 Synthetic division shows that 5 is a zero and g (x) = x – 3 (5 x 2 – 5 x – 20) = (5 x – 3)(x 2 – x – 4). 5 It follows that: f (x) = (2 x + 1) g (x) = (2 x + 1)(5 x – 3)(x 2 – x – 4)

Find zeros when the leading coefficient is not 1 Warm-Up 3 Exercises EXAMPLE STEP 6 Find the remaining zeros of f by solving x 2 – x – 4 = 0. x = – (– 1) + √ (– 1)2 – 4(1)(– 4) 2(1) x = 1 + √ 17 2 Substitute 1 for a, 21 for b, and 24 for c in the quadratic formula. Simplify. ANSWER The real zeros of f are – 1 2 , 3 , 1 + √ 17, and 1 – √ 17. 5 2 2

Warm-Up Exercises GUIDED PRACTICE for Example 3 Find all real zeros of the function. 5. f (x) = 48 x 3+ 4 x 2 – 20 x + 3 ANSWER 3 1 1 4 , 6 , 2

Warm-Up Exercises GUIDED PRACTICE 6. for Example 3 f (x) = 2 x 4 + 5 x 3 – 18 x 2 – 19 x + 42 ANSWER 2, 32 , 1, +2 √ 2

Warm-Up 4 Exercises EXAMPLE Solve a multi-step problem ICE SCULPTURES Some ice sculptures are made by filling a mold with water and then freezing it. You are making such an ice sculpture for a school dance. It is to be shaped like a pyramid with a height that is 1 foot greater than the length of each side of its square base. The volume of the ice sculpture is 4 cubic feet. What are the dimensions of the mold?

Warm-Up 4 Exercises EXAMPLE Solve a multi-step problem SOLUTION STEP 1 Write an equation for the volume of the ice sculpture. 1 4 = 3 x 2(x + 1) 12 = x 3 + x 2 0 = x 3 + x 2 – 12 Write equation. Multiply each side by 3 and simplify. Subtract 12 from each side.

Warm-Up 4 Exercises EXAMPLE Solve a multi-step problem STEP 2 List the possible rational solutions: + 1 , + 2 , + 3 , + 4 , + 6 , +12 1 1 1 STEP 3 Test possible solutions. Only positive xvalues make sense. 1 1 0 – 12 1 2 2 – 10

Warm-Up 4 Exercises EXAMPLE Solve a multi-step problem 2 1 1 0 – 12 1 2 3 6 6 12 0 ↑ 2 is a solution. STEP 4 Check for other solutions. The other two solutions, which satisfy x 2 + 3 x + 6 = 0, are x = – 3 + i √ 15 and can be 2 discarded because they are imaginary numbers.

Solve a multi-step problem Warm-Up 4 Exercises EXAMPLE ANSWER The only reasonable solution is x = 2. The base of the mold is 2 feet by 2 feet. The height of the mold is 2 + 1 = 3 feet.

Warm-Up Exercises GUIDED PRACTICE 7. for Example 4 WHAT IF? In Example 4, suppose the base of the ice sculpture has sides that are 1 foot longer than the height. The volume of the ice sculpture is 6 cubic feet. What are the dimensions of the mold? ANSWER Base side: 3 ft, height: 2 ft

Warm-Up Exercises Daily Homework Quiz 1. List the possible rational zeros of f(x) = x 3 + 8 x 2 – x + 4. ANSWER + 1, + 2, + 4 Find all real zeros of the functions 2. f(x) = x 3 – 3 x 2 – 6 x + 8. ANSWER – 2, 1, 4

Warm-Up Exercises Daily Homework Quiz 3. f(x) = 2 x 3 – 3 x 2 – 17 x + 30. ANSWER – 3, 2, 5 2 4. The volume V of a storage shed with a triangular roof can be modeled by V = x 3 + 1 x 2(6 – x). If 2 the volume of the shed is 80 cubic feet, find x. ANSWER 4