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Difficulty: HardPolynomial Factors and Graphs

The table below shows some values for a cubic polynomial function gg.

xxg(x)g(x)
−1-100
111616
3300

In the xyxy-plane, the graph of y=g(x)y = g(x) is tangent to the xx-axis at x=3x = 3. What is the value of g(0)g(0)?

Answer: 18

Answer

18
The correct value of g(0)g(0) is 18. By using the fact that g(−1)=0g(-1) = 0, we establish (x+1)(x + 1) as a factor. The tangency at x=3x = 3 tells us that (x−3)2(x - 3)^2 is a factor. Writing the function as g(x)=a(x+1)(x−3)2g(x) = a(x + 1)(x - 3)^2 and substituting g(1)=16g(1) = 16 gives 8a=16  ⟹  a=28a = 16 \implies a = 2. Evaluating g(0)g(0) yields 2(1)(9)=182(1)(9) = 18.

Step-by-Step Solution

1
Determine the factors of the cubic polynomial g(x)g(x) using the given zeros and the tangency condition.
The factors are (x+1)(x + 1) and (x−3)2(x - 3)^2, so the function is of the form g(x)=a(x+1)(x−3)2g(x) = a(x + 1)(x - 3)^2.
Since g(−1)=0g(-1) = 0, x=−1x = -1 is a root of the polynomial. The graph being tangent to the xx-axis at x=3x = 3 indicates that x=3x = 3 is a root with a multiplicity of at least 2. Since g(x)g(x) is a cubic polynomial (degree 3), the multiplicity of the root at x=3x = 3 must be exactly 2.
2
Use the table value g(1)=16g(1) = 16 to solve for the constant coefficient aa.
a=2a = 2
Substituting x=1x = 1 into g(x)=a(x+1)(x−3)2g(x) = a(x + 1)(x - 3)^2 gives g(1)=a(1+1)(1−3)2=8ag(1) = a(1 + 1)(1 - 3)^2 = 8a. Setting this equal to the table value of 16 yields 8a=168a = 16, which simplifies to a=2a = 2.
3
Evaluate the polynomial at x=0x = 0 using the fully determined function g(x)=2(x+1)(x−3)2g(x) = 2(x + 1)(x - 3)^2.
18
To find g(0)g(0), substitute x=0x = 0 into the expression: g(0)=2(0+1)(0−3)2=2(1)(9)=18g(0) = 2(0 + 1)(0 - 3)^2 = 2(1)(9) = 18.

Key Concept

Identifying polynomial factors from graphs and tables, and analyzing root multiplicity (tangency vs. crossing).
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Topics

Polynomial Factors and Graphs

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