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Factoring a trinomial











































Explain how to factor the following trinomials forms: x² + bx + c and ax² + bx + c. Is there more than one way to factor this? Show your answer using both words and mathematical notation? Provide an expression for your classmate to factor.

Comments for Factoring a trinomial

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Mar 30, 2013
Factoring
by: Staff


Answer

Part I


A second degree trinomial can be factored as a product of binomials.

There are many different ways to approach this problem.

1. trial and error method
2. quadratic formula
3. decomposition method
3. Triple Play
4. Criss-Cross Method
5. Graphical Method
6. etc.

I am going to cover the "trial and error method" and the "quadratic formula".


trial and error method - works very well in the majority of problems encountered in high school algebra courses

Second Degree Trinomial

Trinomial - 2nd degree trinomial format





Second Degree Trinomial - Example Problem

Trinomial - factoring - example problem



Trial and Error Factoring of Example Problem


1. List all the factors for the "a" term (6, the leading coefficient) and then list all the factor combinations which can be multiplied to = a


Trinomial - factors of 6, the leading coefficient for:  6x² + 4x - 10






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Mar 30, 2013
Factoring
by: Staff


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Part II

2. List all the factors for the "c" term (-10, the constant) and then list all the factor combinations which can be multiplied to = c

Trinomial - factors of the constant term -10 for:  6x² + 4x - 10





3. Write down two sets of empty parentheses

Factoring - begin by writing down two sets of empty parentheses




4. On left side of each set of parentheses, write the variable x.

Factoring - On left side of each set of parentheses, write the variable x.





5. Determine all possible coefficients for the variable x using the factors determined in step 1. There are four possibilities. In each case, when the first terms of each parentheses are multiplied together, the result is 6x². 6x² is the first term of the trinomial we are trying to factor.



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Mar 30, 2013
Factoring
by: Staff


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Part III


Factoring - all possible coefficients for the variable x





5. Determine all possible combinations for the second term within each parentheses using the factors determined in step 2. There are four possible combinations. In each case, when the second terms within each parentheses are multiplied together, the result is -10. -10 is the constant term of the trinomial we are trying to factor.


Factoring - all possible combinations for the second term within each parentheses





6. (a) Synthesize the possible combinations for the second term with the possible combinations for the first term. Altogether there are 16 different possible groupings. (b) Multiply the two binomials for every possibility to see which combinations are factors of the original trinomial.


Binomial Combinations - factoring of trinomial - trial and error 1





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Mar 30, 2013
Factoring
by: Staff


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Part IV


Binomial Combinations - factoring of trinomial - trial and error 2





Binomial Combinations - factoring of trinomial - trial and error 3





Binomial Combinations - factoring of trinomial - trial and error 4






As you can see, there are four combinations which work. (Although it may not look like it, each of these combinations is equivalent to the other.)


Binomial Combinations - factoring of trinomial - trial and error results





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Mar 30, 2013
Factoring
by: Staff


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Part V


A second degree TRINOMIAL can also be factored using the quadratic formula.

The advantage of using this method is that it always works.

The disadvantage of using this method is that the factors will not always be composed of integers.

To use the quadratic equation, rewrite the trinomial as a quadratic equation by setting it equal to zero.


Format of a second degree trinomial:

Trinomial - 2nd degree trinomial format



Format of quadratic equation:
Quadratic Equation - showing the coefficients a, b, and c





Use the Quadratic formula to determine the two factors:

Quadratic formula can be used for factoring a second degree trinomial






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Mar 30, 2013
Factoring
by: Staff


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Part VI


If x1 represents the first solution and x2 represents the second solution, the factors are:

Factors of a second degree trinomial using the quadratic formula




When the quadratic formula is applied to the example problem:

a = 6
b = 4
c = -10


Quadratic Formula applied to the example problem:  6x² + 4x -10





Calculation of x_1 and x_2 using the quadratic formula




The binomial factors are:

Factors using the quadratic formula results





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Mar 30, 2013
Factoring
by: Staff


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Part VII

But, . . . why doesn't multiplying the factors equal the original trinomial?

If you multiply each side of the equation by 6, the original trinomial in the example appears.

Quadratic formula factors reconciled with original trinomial








Thanks for writing.

Staff
www.solving-math-problems.com


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