Problem of the Week

Updated at Nov 6, 2023 1:33 PM

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Mar 24, 2025

This week we have another algebra problem:

How can we factor $36{v}^{2}-66v+30$ ?

Let's start!

36{v}^{2}-66v+30

Find the Greatest Common Factor (GCF).

What is the largest number that divides evenly into

36{v}^{2}

-66v

, and

30

It is

6

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

36{v}^{2}

-66v

, and

30

It is 1, since

v

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{36{v}^{2}}{6}+\frac{-66v}{6}+\frac{30}{6})

Simplify each term in parentheses.

6(6{v}^{2}-11v+5)

Split the second term in

6{v}^{2}-11v+5

into two terms.

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

6\times 5=30

Ask: Which two numbers add up to

-11

and multiply to

30

-5

and

-6

Split

-11v

as the sum of

-5v

and

-6v

6{v}^{2}-5v-6v+5

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6(6{v}^{2}-5v-6v+5)

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

6(v(6v-5)-(6v-5))

Factor out the common term

6v-5

6(6v-5)(v-1)

Done