Question

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Given the expression: 5x^10 − 80x^2

Part A: Rewrite the expression by factoring out the greatest common factor. (4 points)

Part B:Factor the entire expression completely. Show the steps of your work. (6 points)

Answer 1

5x² ( x² - 2x + 2 ) ( x² + 2x + 2 ) ( x² + 2 ) ( x² - 2 )

Explanation:

→ Find the highest common factor from 5 and 80

5

→ Find the highest common factor of x¹⁰ and x²

x²

→ Write the partially factored form

5x² ( x⁸ - 16 )

→ Factor down ( x⁸ - 16 )

( x⁴ + 4 ) ( x⁴ - 4 )

→ Factor down ( x⁴ + 4 ) ( x⁴ - 4 )

( x² - 2x + 2 ) ( x² + 2x + 2 ) ( x² + 2 ) ( x² - 2 )

Answer 2

• 5x^2(x^8 -16)
• 5x^2(x^2 +2x +2)(x^2 -2x +2)(x^2 +2)(x^2 -2)

Explanation:

The greatest common factor is that factor that includes every prime factor that is found in both expressions. Once that is removed, the formula for the factorization of the difference of squares is needed.

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A

Each expression factors as ...

5x^10 = 5·x^2·x^8

80x^2 = 5·x^2·16

The common factors in these expressions are 5, x^2, so the expression with the greatest common factor factored out is ...

5x^2(x^8 -16)

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B

The remaining factor is the difference of squares, which factors as ...

a² -b² = (a -b)(a +b)

Here, we have a=x^4 and b=2^2, so the factors of the difference are ...

x^8 -16 = (x^4 -4)(x^4 +4)

The first of these factors is also recognizable as a difference of squares, so it can be factored further:

x^4 -4 = (x^2 -2)(x^2 +2)

The second factor can also be cast as the difference of squares, allowing it to be factored further:

x^4 +4 = (x^4 +4x^2 +4) -(4x^2) = (x^2 +2)^ -(2x)^2

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

So, the complete factorization of the original expression is ...

5x^10 -80x^2 = 5x^2(x^2 +2x +2)(x^2 -2x +2)(x^2 +2)(x^2 -2)

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Additional comment

We could factor to linear terms if we wanted to use radicals and complex numbers. This is the factorization in integers.