Problem of the Week

Updated at Jan 15, 2024 3:07 PM

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Mar 24, 2025

How would you find the factors of $30{x}^{2}-57x+21$ ?

Below is the solution.

30{x}^{2}-57x+21

Find the Greatest Common Factor (GCF).

What is the largest number that divides evenly into

30{x}^{2}

-57x

, and

21

It is

3

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

30{x}^{2}

-57x

, and

21

It is 1, since

x

is not in every term.

Multiplying the results above,

The GCF is

3

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GCF =

3

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

3(\frac{30{x}^{2}}{3}+\frac{-57x}{3}+\frac{21}{3})

Simplify each term in parentheses.

3(10{x}^{2}-19x+7)

Split the second term in

10{x}^{2}-19x+7

into two terms.

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

10\times 7=70

Ask: Which two numbers add up to

-19

and multiply to

70

-5

and

-14

Split

-19x

as the sum of

-5x

and

-14x

10{x}^{2}-5x-14x+7

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3(10{x}^{2}-5x-14x+7)

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

3(5x(2x-1)-7(2x-1))

Factor out the common term

2x-1

3(2x-1)(5x-7)

Done