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Question 28 of 30

beginning of content:

M:C:MC:21

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SAT

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The function f is defined by f open parentheses x close parentheses equals 2 x cubed plus 3 x squared plus c x plus 8 comma where c is a constant. In the xy-plane, the graph of f intersects the x-axis at the three points (-4,0), open parentheses 1 half comma space 0 close parentheses comma and left parenthesis p comma space 0 right parenthesis. What is the value of c?

Select an Answer

-18

Correct Answer: 
Yes

-2

Correct Answer: 
No

2

Correct Answer: 
No

10

Correct Answer: 
No

View Correct Answer

Choice A is correct. The given zeros can be used to set up an equation to solve for c. Substituting –4 for x and 0 for y yields minus 4 c equals 72 comma or c equals minus 18. Alternatively, since –4, 1 half comma and p are zeros of the polynomial function f left parenthesis x right parenthesis equals 2 x cubed plus 3 x squared plus c x plus 8 comma it follows that f left parenthesis x right parenthesis equals left parenthesis 2 x minus 1 right parenthesis left parenthesis x space plus 4 right parenthesis left parenthesis x minus p right parenthesis. Were this polynomial multiplied out, the constant term would be left parenthesis minus 1 right parenthesis left parenthesis 4 right parenthesis left parenthesis minus p right parenthesis equals 4 p. (We can see this without performing the full expansion.) Since it is given that this value is 8, it follows that 4 p equals 8 or rather, p equals 2. Substituting 2 for p in the polynomial function yields f left parenthesis x right parenthesis equals left parenthesis 2 x minus 1 right parenthesis left parenthesis x plus 4 right parenthesis left parenthesis x minus 2 right parenthesis comma and after multiplying the factors one finds that the coefficient of the x term, or the value of c, is –18.

Choice B is not the correct answer. This value is a misunderstood version of the value of p, not c, and the relationship between the zero and the factor (if a is the zero of a polynomial, its corresponding factor is x minus a) has been confused.

Choice C is not the correct answer. This is the value of p, not c. Using this value as the third factor of the polynomial will reveal that the value of c is -18.

Choice D is not the correct answer. This represents a sign error in the final step in determining the value of c.

Question Difficulty: 
hard
Objective: 

Students must understand the zeros of a polynomial function and how they are used to construct algebraic representations of polynomials.