### Factoring Monomials Worksheets

The factor form of 2x + 2 is 2(x+1) since is common to both terms. This example shows us what factoring really means. Factoring is just writing a polynomial as the product of a monomial and simpler polynomial or other polynomials. Same as multiplying monomial to a polynomial, it also uses the distributive property but it “undoes” or reverses multiplying.

Factoring monomials from polynomials can be done through the following:

1. Factor out the Greatest common factor (GCF) – the largest term that can be factored out from all the terms in the polynomial. The GCF of the numerical part is the largest number that can be used as the divisor of the numbers found in the expression such as 12 and 36 which can be both divided by 12, thus the GCF of these two numbers is 12. When there’s a common variable, use the least exponent that appears on that variable as the factor. Moreover, when factoring apply the quotient rule for exponents which implies that $$\frac{a^m}{a^n}=a^{m-n}$$. Here are examples of Factoring out the GCF.

1.1 Factor $$8x+12$$.

Solution:

\small \begin{align*} & 8x+12 \!\!\!\!&=&\; 4\left ( 2x \right )+4\left ( 3 \right ) && \textup{Factor out 4 in each term} \\ &&=& \; 4\left ( 2x+3 \right )&& \textup{Since GCF is 4, factor out 4 }\\ \end{align*}

1.2 Factor $$\left (9x^2+12x^3\right)$$.

Solution:
\small \begin{align*} & 9x^2+12x^3 \!\!\!\!&=&\; 3x^2\left ( 3 \right )+3x^2\left ( 4x \right ) \\ &&=& \; 3x^2\left ( 3+4x \right )\\ \end{align*}
The greatest factor of numbers 3 and 9 is 3, while the least exponent that appends on the variable $$x$$ is 2. Thus, the GCF is $$3x^2$$.

1.3 Factor $$32p^4-24p^3+16p^5$$.

$32p^4-24p^3+16p^5=8p^3 \left(4p-3+2p^2\right)$

1.4 Factor $$24m^3n^2-18m^2n+6m^4n^3$$.

Solution:
\small \begin{align*} & 24m^3n^2-18m^2n+6m^4n^3 \!\!\!\!&=&\; 6m^2n\left ( 4mn \right )-6m^2n\left ( 3 \right )-6m^2n\left ( m^2n^2 \right ) \\ &&=& \; 6m^2n\left ( 4mn-3+m^2n^2 \right )\\ \end{align*}
The largest divisor of numbers 24, 18 and 6 is 6. While the least exponent that appears on the variable $$m$$ is 2 and in variable $$n$$ is 1, thus the GCF is $$6m^2n$$.

2. Factoring by grouping – this can be done through grouping terms, factoring out the common monomial factor in each group and by factoring out the entire polynomial if there’s a common factor. This will commonly result to multiplying two binomials. You can check if your answer is correct by multiplying your final factors.

Examples:

2.1 Factor $$ax+by+bx+by$$.

Solution:
\small \begin{align*} & ax+by+bx+by \!\!\!\!&=&\; \left ( ax+ay \right )+\left ( bx+by \right ) && \textup{Group terms} \\ &&=& \; a\left ( x+y \right )+b\left ( x+y \right )&& \textup{Factor out respective common terms }\\&&=&\;\left ( x+y \right )\left ( a+b \right )&& \textup{Final factored form} \end{align*}

2.2 Factor $$3x-3y-ax+ay$$.

Solution:
\small \begin{align*} & 3x-3y-ax+ay \!\!\!\!&=&\; \left (3x-3y \right )+\left (-ax-ay \right ) \\ &&=& \; 3\left ( x-y\right )-a\left ( x-y \right )\\&&=& \;\left (x-y \right ) \left (3-a \right )\end{align*}

If the given is not arrange from the highest degree to the lowest, we need to rearrange it before grouping terms. As an example look at the following:

2.3 Factor $$p^2q^2-10-2q^2+5p^2$$.

Solution:
\small \begin{align*} & p^2q^2-10-2q^2+5p^2 \!\!\!\!&=&\; p^2q^2-2q^2+5p^2-10 &&\textup{Rearrange}\\ &&=& \; \left (p^2q^2-2q^2\right )+\left ( 5p^2-10 \right )&&\textup{Group terms}\\&&=& \;q^2\left (p^2-2 \right )+5 \left (p^2-2 \right )&&\textup{Factoring out common factor}\\&&=& \;\left ( q^2+5 \right )\left (p^2-2 \right ) &&\textup{Final factored form}\end{align*}

2.4 Factor $$8+9y^4-6y^3-12y$$.

Solution:
\small \begin{align*} & 8+9y^4-6y^3-12y \!\!\!\!&=&\; 9y^4-6y^3-12y+8 \\ &&=& \; \left (9y^4-6y^3\right )+\left ( -12y+8 \right )\\&&=& \;3y^3\left (3y-2 \right )-4 \left (3y-2 \right )\\&&=& \;\left ( 3y-2 \right )\left (3y^3 -4 \right ) \end{align*}
To check:
\small \begin{align*} & \left (3y-2 \right )\left (3y^3-4 \right ) \!\!\!\!&=&\; \left (3y\right )\left (3y^3 \right )+\left ( -2 \right )\left ( -4 \right )+\left (-2 \right )\left (3y^3 \right )+\left ( 3y \right )\left (-4 \right ) \\ &&=& \; 9y^4+8-6y^3-12y\\&&=&\; 9y^4-6y^3-12y+8 \end{align*}

Practice Exercises:

Factor out the greatest common factor and simplify the factors, if possible.
1. $$12m+60$$
2. $$15a^2c^3-25ac^2+5ac$$
3. $$6k^3+36k^4-48k^5$$
4. $$12k^3m+24k^3m^2+36k^8$$
5. $$xy-5xy^2$$