Problem of the Week

Updated at Dec 18, 2017 3:47 PM

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May 5, 2025

How can we compute the factors of $18{x}^{2}-24x+6$ ?

Below is the solution.

18{x}^{2}-24x+6

Find the Greatest Common Factor (GCF).

What is the largest number that divides evenly into

18{x}^{2}

-24x

, and

6

It is

6

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

18{x}^{2}

-24x

, and

6

It is 1, since

x

is not in every term.

Multiplying the results above,

The GCF is

6

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GCF =

6

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

6(\frac{18{x}^{2}}{6}+\frac{-24x}{6}+\frac{6}{6})

Simplify each term in parentheses.

6(3{x}^{2}-4x+1)

Split the second term in

3{x}^{2}-4x+1

into two terms.

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

3\times 1=3

Ask: Which two numbers add up to

-4

and multiply to

3

-1

and

-3

Split

-4x

as the sum of

-x

and

-3x

3{x}^{2}-x-3x+1

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6(3{x}^{2}-x-3x+1)

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

6(x(3x-1)-(3x-1))

Factor out the common term

3x-1

6(3x-1)(x-1)

Done