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Factorization of Quadratic Expressions with Negative Coefficients

Factor quadratics with positive and negative coefficients

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Factorization of Quadratic Expressions with Negative Coefficients

Factorization of Quadratic Expressions with Negative Coefficients

Factor a quadratic trinomial when a = 1, b is negative and c is positive

Now let’s see how this method works if the middle coefficient is negative.

Factor .

We are looking for an answer that is a product of two binomials in parentheses:

When negative coefficients are involved, we have to remember that negative factors may be involved also. The number 8 can be written as the product of the following numbers:

but also

and

but also

The last option is the correct choice. The answer is . We can check to see if this is correct by multiplying :

The answer checks out.

Factoring

1. Factor .

We are looking for an answer that is a product of two binomials in parentheses:

The number 16 can be written as the product of the following numbers:

The answer is .

In general, whenever is negative and and are positive, the two binomial factors will have minus signs instead of plus signs.

Factor when a = 1 and c is Negative

Now let’s see how this method works if the constant term is negative.

2. Factor .

We are looking for an answer that is a product of two binomials in parentheses:

Once again, we must take the negative sign into account. The number -15 can be written as the product of the following numbers:

The answer is .

We can check to see if this is correct by multiplying:

The answer checks out.

3. Factor .

We are looking for an answer that is a product of two binomials in parentheses:

The number -24 can be written as the product of the following numbers:

The answer is .

Factor when a = - 1

When , the best strategy is to factor the common factor of -1 from all the terms in the quadratic polynomial and then apply the methods you learned so far in this section

4. Factor .

First factor the common factor of -1 from each term in the trinomial. Factoring -1 just changes the signs of each term in the expression:

We’re looking for a product of two binomials in parentheses:

Now our job is to factor .

The number -6 can be written as the product of the following numbers:

The answer is .

Example

Example 1

Factor .

We are looking for an answer that is a product of two binomials in parentheses:

The number -35 can be written as the product of the following numbers:

The answer is .

Review 

Factor the following quadratic polynomials.

Review (Answers)

To view the Review answers, open this PDF file and look for section 9.9. 

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Vocabulary

TermDefinition
Quadratic Polynomials A quadratic polynomial is a polynomial of the 2nd degree, in other words, a polynomial with an x^2 term.

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