Problem of the Week

Updated at Aug 7, 2017 12:35 PM

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Apr 28, 2025

This week's problem comes from the algebra category.

How can we compute the factors of $10{x}^{2}-24x+8$ ?

Let's begin!

10{x}^{2}-24x+8

Find the Greatest Common Factor (GCF).

What is the largest number that divides evenly into

10{x}^{2}

-24x

, and

8

It is

2

What is the highest degree of $x$ that divides evenly into

10{x}^{2}

-24x

, and

8

It is 1, since

x

is not in every term.

Multiplying the results above,

The GCF is

2

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GCF =

2

Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)

2(\frac{10{x}^{2}}{2}+\frac{-24x}{2}+\frac{8}{2})

Simplify each term in parentheses.

2(5{x}^{2}-12x+4)

Split the second term in

5{x}^{2}-12x+4

into two terms.

Multiply the coefficient of the first term by the constant term.

5\times 4=20

Ask: Which two numbers add up to

-12

and multiply to

20

-2

and

-10

Split

-12x

as the sum of

-2x

and

-10x

5{x}^{2}-2x-10x+4

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2(5{x}^{2}-2x-10x+4)

Factor out common terms in the first two terms, then in the last two terms.

2(x(5x-2)-2(5x-2))

Factor out the common term

5x-2

2(5x-2)(x-2)

Done