Question

(08.01, 8.02 MC)

An expression is shown below:
4x3 - 20x2 +36x - 180
Part A: Rewrite the expression so that the GCF is factored completely. (4 points)
Part B: Rewrite the expression completely factored. Show the steps of your work. (6
points)
(10 points)

Answer 1

• 4(x^3 -5x^2 +9x -45)
• 4(x -5)(x^2 +9) = 4(x -5)(x -3i)(x +3i)

Explanation:

A) We observe that each coefficient is divisible by 4. The GCF of the terms is 4.

4x^3 - 20x^2 +36x - 180 = 4(x^3 -5x^2 +9x -45)

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B) We further observe that the coefficients of each pair are in the ratio 1:-5, so we can factor pairwise:

= 4(x^2(x -5) +9(x -5)) = 4(x^2 +9)(x -5) . . . . factored completely in reals

The quadratic factor x^2+9 can be factored further if complex numbers are used:

= 4(x -3i)(x +3i)(x -5) . . . . factored completely in complex numbers

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Comment on the quadratic factor

The term x^2 +9 can be considered to be the difference of squares:

x^2 +9 = x^2 -(-9)

= x^2 -(3i)^2

= (x -3i)(x +3i)