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Factoring When the Leading Coefficient Doesn't Equal 1

The area of a square is . What are the dimensions of the square?

When the  term, the coefficient of the term in a quadratic function in the form  is greater than one, factoring it may be a bit tricky. This is because the  term is the result of multiplying the coefficients of the  terms from the source binomials, so you need to identify the different coefficients that are factors of

As an introduction, consider FOIL-ing two binomials when the coefficients in front of the terms are not 1:

Multiply, using the FOIL method: .

FIRST:

OUTSIDE:

INSIDE:

LAST:

Combining all the terms together, you get:

Now, try working backwards and factor a trinomial with  to get two factors. Remember, you can always check your work by multiplying the final factors together.

Factor

This is a factorable trinomial. When there is a coefficient, or number in front of, , you must follow all the steps you are familiar with to factor. Multiply together and  and then find the two numbers whose product is and sum is One way to organize this information is to draw a large X with  in the top opening, and  in the bottom one:

For the side sections, you are looking for the two factors of  (12, in this case) that sum to be equal to

Factors Sum
-1, 12 11
1, -12 -11
2, -6 -4
-2, 6 4
-3, 4 1

The factors that work are 3 and -4. Now, take these factors and rewrite the  term, expanded as

Next, group the first two terms together and the last two terms together and pull out any common factors.

What is in the parenthesis is the same. We now have two terms that both have as factor. Pull this factor out.

The factors of are . You can FOIL these to check your answer.

Now, let's factor .

Let’s make the steps we followed in the previous problem a little more concise.

Step 1: Find and the factors of this number that add up to .

The factors of -20 that add up to 8 are 10 and -2.

Step 2: Rewrite the trinomial with the term expanded, using the two factors from Step 1.

Step 3: Group the first two and second two terms together, find the GCF and factor again.

Alternate Method: What happens if we list before in Step 2?

This tells us it does not matter which term we list first in Step 2 above.

Let's factor .

Let’s use the steps from the previous problem, but we are going to add an additional step at the beginning.

Step 1: Look for any common factors. Pull out the GCF of all three terms, if there is one.

This will make it much easier for you to factor what is inside the parenthesis.

Step 2: Using what is inside the parenthesis, find and determine the factors that add up to

The factors of -60 that add up to -11 are -15 and 4.

Step 3: Rewrite the trinomial with the term expanded, using the two factors from Step 2.

Step 4: Group the first two and second two terms together, find the GCF and factor again.

Examples

Example 1

Earlier, you were asked to find the dimensions of the square.

The dimensions of a square are its length and its width, so we need to factor the area .

We need to multiply together and (from ) and then find the two numbers whose product is and whose sum is .

Now we can see that we need the two factors of 144 that also add up to 24. Testing the possibilities, we find that and .

Now, take these factors and rewrite the term expanded using 12 and 12.

Next, group the first two terms together and the last two terms together and pull out any common factors.

We now have two terms that both have as factor. Pull this factor out.

The factors of are , which are also the dimensions of the square.

Example 2

Multiply .

FOIL:

Factor the following quadratics, if possible.

Example 3

Use the steps from the examples above. There is no GCF, so we can find the factors of that add up to .

The factors of -45 that add up to -4 are -9 and 5.

Example 4

has a GCF of 3. Pulling this out, we have . There is no number in front of , so we see if there are any factors of -6 that add up to 2. There are not, so this trinomial is not factorable.

Example 5

also has a GCF of 3. Pulling this out, we have . . The factors of -24 than add up to -10 are -12 and 2.

Example 6

.  has a GCF of 4. Pulling this out, we have . This trinomial does not have a number in front of , so we can use the shortcut we are familiar with. What are the factors of -12 that add up to 1?

Review

Multiply the following expressions.

Factor the following quadratic equations, if possible. If they cannot be factored, write not factorable. Don’t forget to look for any GCFs first.

1. Factor . What is ?
2. Factor . What is ? What types of numbers are and ?

To see the Review answers, open this PDF file and look for section 5.2.

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
Greatest Common Factor The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.
linear factors Linear factors are expressions of the form $ax+b$ where $a$ and $b$ are real numbers.
Polynomial A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
Quadratic Formula The quadratic formula states that for any quadratic equation in the form $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
Quadratic Polynomials A quadratic polynomial is a polynomial of the 2nd degree, in other words, a polynomial with an $x^2$ term.
Trinomial A trinomial is a mathematical expression with three terms.