### Common factor only

You already know how to use the distributive property of multiplication to multiply a monomial by a polynomial.  For example, you can easily perform the following multiplication:

$$2x(3x + 4y) = 6x^2 + 8xy$$

But is there a way to begin with the polynomial, and break it down into its original factors?  That is, could we look at a polynomial like $$6x^2 + 8xy$$ and figure out that it is equal to $$2x(3x + 4y)$$?. The answer is yes.  We will use a method called factoring.

In this article we will learn to factor a polynomial by searching for the greatest common factor of all of the terms in the polynomial.  Let’s get right to it.  Consider the polynomial

$$80v^2u - 8v^3 + 40v^2$$

We wish to find the greatest common factor of the three different terms above.  Let’s consider them one at a time.

By breaking $$80v^2u$$ into its prime factorization, we find that

$$80v^2u = 2\cdot 2\cdot 2\cdot 2\cdot 5\cdot v\cdot v\cdot u$$

Similarly, we look at $$-8v^3$$ and we find

$$-8v^3 = -1\cdot 2\cdot 2\cdot 2\cdot v\cdot v\cdot v$$

Finally, we break down $$40v^2$$ into

$$40v^2 = 2\cdot 2\cdot 2\cdot 5\cdot v\cdot v$$

Now we pick out the numbers that are common to every single term.  We see that $$2$$ appears in each term three times.  So we will be able to factor out $$2\cdot 2\cdot 2 = 8$$ from the polynomial.  Also, we see that each and every term contains $$v\cdot v = v^2$$. So we factor it out.  Then

$$80v^2u - 8v^3 + 40v^3 = 8v^2$$(???)

Well, we have factored out the greatest common factor $$8v^2$$. Now what is left of the polynomial inside the parentheses?  Whatever is left from each term!  Let’s work it out.

When we factor out $$8v^2$$ from $$80v^2u$$, we are left with $$2\cdot 5\cdot u = 10u$$. Then $$10u$$ will be inside the parentheses.

Similarly, when we factor out $$8v^2$$ from $$-8v^3$$, we are left with $$-v$$.

Finally, when we factor out $$8v^2$$ from $$40v^2$$,  we are left with $$5$$.

Then our final factorization is $$80v^2u - 8v^3 + 40v^2 = 8v^2(10u - v + 5)$$. Congratulation, you have just factored!

Let’s quickly try another example.  Factor the polynomial $$-12x + 6xy^2 - 15x^3y^3$$. To do this we consider each term separately and determine the greatest common factor.  Can you figure out what it is?  The greatest common factor is $$3x$$! !  If you cannot see this, repeat the procedure we did above for finding the greatest common factor.  Well what happens when we factor out $$3x$$ from each term?  The first term is left with $$-4$$, the second term is left with $$2y^2$$, and the third term is left with $$-5x^2y^3$$. Then our final factorization is

$$-12x + 6xy^2 - 15x^3y^3 = 3x(-4 + 2y^2 - 5x^2y^3)$$

10424 x

Factor the common factor out of each expression.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

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7658 x

Factor the common factor out of each expression.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Watch bellow how to solve this example:

5619 x

Factor the common factor out of each expression.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Watch bellow how to solve this example:

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