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Difficulty: HardPolynomial Factors and Graphs

In the xyxy-plane, the graph of the cubic function f(x)=x3−7x2+kx−12f(x) = x^3 - 7x^2 + kx - 12, where kk is a constant, is tangent to the xx-axis at one point and intersects the xx-axis at another point. If all roots of f(x)f(x) are real numbers, which of the following could be the value of kk?

  1. 16Answer
  2. B
    -16
  3. C
    7
  4. D
    12

Answer

16
The correct answer is the value that makes the cubic function have a double root at 22 and a single root at 33. Setting f(x)=(x−2)2(x−3)f(x) = (x - 2)^2(x - 3) and expanding it yields x3−7x2+16x−12x^3 - 7x^2 + 16x - 12. Comparing this to f(x)=x3−7x2+kx−12f(x) = x^3 - 7x^2 + kx - 12 shows that k=16k = 16.

Step-by-Step Solution

1
Express the cubic function in factored form based on its roots.
f(x)=(x−r1)2(x−r2)f(x) = (x - r_1)^2(x - r_2)
Since the graph is tangent to the xx-axis at one point and intersects it at another, it has one real root of multiplicity 2 (let it be r1r_1) and one real root of multiplicity 1 (let it be r2r_2).
2
Expand the factored form to obtain coefficients in terms of r1r_1 and r2r_2.
f(x)=x3−(2r1+r2)x2+(r12+2r1r2)x−r12r2f(x) = x^3 - (2r_1 + r_2)x^2 + (r_1^2 + 2r_1 r_2)x - r_1^2 r_2
Comparing this expansion to the given expression f(x)=x3−7x2+kx−12f(x) = x^3 - 7x^2 + kx - 12 allows us to set up a system of equations.
3
Set up and solve the system of equations for r1r_1 and r2r_2.
2r1+r2=72r_1 + r_2 = 7, r12r2=12r_1^2 r_2 = 12, and k=r12+2r1r2k = r_1^2 + 2r_1 r_2. Substituting r2=7−2r1r_2 = 7 - 2r_1 into the second equation gives 2r13−7r12+12=02r_1^3 - 7r_1^2 + 12 = 0. Testing integer root candidates shows r1=2r_1 = 2 is a solution, which yields r2=3r_2 = 3.
This determines the values of the roots r1r_1 and r2r_2.
4
Calculate the value of kk using the values of the roots.
k=22+2(2)(3)=16k = 2^2 + 2(2)(3) = 16
Substituting r1=2r_1 = 2 and r2=3r_2 = 3 into the expression for kk gives the target coefficient.

Key Concept

Analyzing the relationship between a polynomial's algebraic factors, roots, and its graphical features such as tangency and intercepts.
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Topics

Polynomial Factors and Graphs

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