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Factorization of Quadratic Expressions

Factor quadratics with positive coefficients

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Factorization of Quadratic Expressions

What if you had a quadratic expression like in which all the coefficients were positive? How could you factor that expression? After completing this Concept, you'll be able to factor quadratic expressions like this one with positive coefficient values.

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CK-12 Foundation: 0908S Factoring Quadratic Expressions

Guidance

Quadratic polynomials are polynomials of the degree. The standard form of a quadratic polynomial is written as

where and stand for constant numbers. Factoring these polynomials depends on the values of these constants. In this section we’ll learn how to factor quadratic polynomials for different values of and . (When none of the coefficients are zero, these expressions are also called quadratic trinomials, since they are polynomials with three terms.)

You’ve already learned how to factor quadratic polynomials where . For example, for the quadratic , the common factor is and this expression is factored as . Now we’ll see how to factor quadratics where is nonzero.

Factor when a = 1, b is Positive, and c is Positive

First, let’s consider the case where is positive and is positive. The quadratic trinomials will take the form

You know from multiplying binomials that when you multiply two factors , you get a quadratic polynomial. Let’s look at this process in more detail. First we use distribution:

Then we simplify by combining the like terms in the middle. We get:

So to factor a quadratic, we just need to do this process in reverse.

This means that we need to find two numbers and where

The factors of are always two binomials

such that and .

Example A

Factor .

Solution

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

We want two numbers and that multiply to 6 and add up to 5. A good strategy is to list the possible ways we can multiply two numbers to get 6 and then see which of these numbers add up to 5:

So the answer is .

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

The answer checks out.

Example B

Factor .

Solution

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

The number 12 can be written as the product of the following numbers:

The answer is .

Example C

Factor .

Solution

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

The number 12 can be written as the product of the following numbers:

The answer is .

Watch this video for help with the Examples above.

CK-12 Foundation: Factoring Quadratic Expressions

Vocabulary

  • A quadratic of the form factors as a product of two binomials in parentheses:
  • If and are positive, then both and are positive.

Guided Practice

Factor .

Solution

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

The number 36 can be written as the product of the following numbers:

The answer is .

Practice

Factor the following quadratic polynomials.

Vocabulary

Greatest Common Factor

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

Linear factors are expressions of the form ax+b where a and b are real numbers.
Polynomial

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

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

Quadratic Polynomials

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

Trinomial

A trinomial is a mathematical expression with three terms.

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