Notes on Factoring Quadratic Expressions



Notes on Factoring Trinomials - Page I Name_________________________

MM1A2. Students will simplify and operate with radical expressions, polynomials,

and rational expressions.

f. Factor expressions by greatest common factor, grouping, trial and error, and

special products limited to the formulas below.

Before learning how to factor polynomials with three terms, called trinomials, three important skills must be mastered.

First, it is important to learn how to write an algebraic expression in standard form. An expression is in standard form if the terms are arranged from the highest to the lowest exponent of the chosen variable.

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Consider this algebraic expression: [pic].

Since there is only one variable, the expression should be arranged from the highest to the lowest exponent of the variable x. The exponents of x are 1, 3, 5, 6, and 0.

In standard form, the expression reads [pic].

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For Questions 1-2, rewrite the expression in standard form.

1. [pic] 2. [pic]

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The second skill required is to be able to find a pair of unique integers that have a given product and a given sum.

Find two numbers with the given product and sum.

3. product: 15 4. product: 18 5. product: 48

sum: 8 sum: 9 sum: 16

numbers: 3 & 5

6. product: -10 7. product: -32 8. product: 21

sum: 3 sum: -4 sum: -10

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The third skill is recognizing a quadratic expression. A quadratic expression is any algebraic expression that can be written in the form

[pic].

An example of a quadratic expression is [pic].

The letters a, b, and c can all be any real number (as long as [pic]), and the variable x is arbitrary.

Note that the highest exponent for a variable in a quadratic expression is 2.

For example, the expression [pic] is a quadratic expression. Note how one can find the values of a, b, and c in this expression:

[pic]

[pic]

The value of a is 5, the value of b is -10, and the value of c is -2. It's important to observe that these values are negative if there is a subtraction sign in front of the term.

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Consider [pic].

This expression, when rewritten in standard form, is [pic].

This is a quadratic expression with the following values: [pic], [pic], and [pic].

Note that a = 1 because the understood coefficient of [pic] is 1 because [pic].

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For Questions 9-10, is the given expression a quadratic expression? If so, rewrite it in standard form, if necessary. In addition, state the values of a, b, and c.

9. [pic] 10. [pic] 11. [pic]

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How to Factor a Trinomial:

Step 1 – Rewrite it in standard form (if necessary).

Step 2 - Factor out the GCF (if necessary). It goes along for the ride.

If the remaining expression is a quadratic trinomial, and a = 1,

Step 3 – Find two numbers, #1 and #2, with a product of c and a sum of b. If two numbers do

not exist, then the expression at the end of the previous step is the answer.

Step 4 - Rewrite the expression in factored form as [pic].

Obviously again, the use of the variable x depends on the expression - it is arbitrary.

Also, remember that "factoring" an expression is essentially the opposite of the processes known as "distributing", "simplifying", or "multiplying".

Finally, it must emphasized again that in order to do steps 3 & 4, the expression must be quadratic, it must be a trinomial, and the value of a must be 1.

Notes on Factoring Trinomials - Page II Name_________________________

Factor [pic].

First, it's important to note that this is a quadratic trinomial because the highest exponent is two, and there are three terms. Also, note that a = 1.

Step 1: This step can be ignored because the expression is already in standard form.

Step 2: This step can also be ignored because the GCF is just 1.

Step 3: In this expression, [pic] and [pic]. Thus, we need two numbers with a product

of -22 and a sum of -9. Those two numbers are 2 and -11.

Step 4: We now rewrite the expression as [pic].

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Factor [pic].

It is important to know that the variable x has been replaced by s --- no big deal. Also, this is a quadratic trinomial again.

Step 1: When rewritten in standard form, the expression reads [pic].

Step 2: Again, this step can be skipped, because the GCF is 1.

Step 3: In this expression, [pic] and [pic]. Thus, we need two numbers with a product

of 40 and a sum of -14. Those two numbers are -4 and -10.

Step 4: We now rewrite the expression as [pic].

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Factor the given expression.

12. [pic] 13. [pic]

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Factor [pic].

Step 1: The expression is in standard form.

Step 2: There is a GCF of 2r, so when it has been factored out, the expression reads

[pic].

Step 3: The remaining expression is a quadratic trinomial in which a = 1. In this expression,

[pic] and [pic]. Thus, we need two numbers with a product of 16 and a sum of -8.

Those two numbers are -4 and -4.

Step 4: We now rewrite the expression as [pic]. It is vitally important that one not

forget the GCF. "It goes along for the ride."

14. Factor [pic]. 15. Factor [pic].

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Homework on Factoring Trinomials

Factor the expression. If nothing can be factored, write PRIME.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11. [pic] 12. [pic]

13. [pic] 14. [pic]

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1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. PRIME

9. [pic] 10. [pic] 11. [pic]

12. PRIME - it is not quadratic because the highest exponent is 3, not 2.

13. [pic] 14. [pic]

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[pic]

[pic]

[pic]

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