Question

select the correct answer
factor each polynomial function. based on these factors, which polynomials graph crosses the x-axis at only one point?
a. $f(x)=x^3 - 2x^2 - 16x + 32$
b. $f(x)=x^3 - 7x^2 + 3x - 21$
c. $f(x)=x^3 - 3x^2 - 4x + 12$
d. $f(x)=x^3 + 2x^2 - 9x - 18$

Explanation:

Step1: Factor Option A by grouping

Group terms: $(x^3-2x^2)+(-16x+32)$
Factor: $x^2(x-2)-16(x-2)=(x-2)(x^2-16)=(x-2)(x-4)(x+4)$
Roots: $x=2,4,-4$ (3 distinct x-intercepts)

Step2: Factor Option B by grouping

Group terms: $(x^3-7x^2)+(3x-21)$
Factor: $x^2(x-7)+3(x-7)=(x-7)(x^2+3)$
Roots: $x=7$ (real root; $x^2+3=0$ has no real roots)

Step3: Factor Option C by grouping

Group terms: $(x^3-3x^2)+(-4x+12)$
Factor: $x^2(x-3)-4(x-3)=(x-3)(x^2-4)=(x-3)(x-2)(x+2)$
Roots: $x=3,2,-2$ (3 distinct x-intercepts)

Step4: Factor Option D by grouping

Group terms: $(x^3+2x^2)+(-9x-18)$
Factor: $x^2(x+2)-9(x+2)=(x+2)(x^2-9)=(x+2)(x-3)(x+3)$
Roots: $x=-2,3,-3$ (3 distinct x-intercepts)

Answer:

B. $f(x) = x^3 - 7x^2 + 3x - 21$