factoring or writing a polynomial as a product can be useful for finding the x-intercepts or zeros....
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Factoring or writing a polynomial as a product can be useful for finding the x-intercepts or zeros.
Solving a quadratic equation means finding the zeros or x-intercepts.
We’ll now look at finding zeros by factoring.
6-32
First let’s review how to multiply polynomials using the Distributive Law, given an equation in the form
y = (x + 4)(x - 1).
To multiply two linear factors, you apply the Distributive Law twice. So, y = (x + 4)(x - 1) becomes
2
1 4 1
4 4
y x x x
y x x x
and combine like terms
2 3 4y x x
The equation is now written in standard form. To see this in another way, let’s look at y = (x + 4) (x - 1) using a multiplication table.
2 3 4y x x
1. In the left-hand column, write the terms of the first factor, x and 4.
2. In the first row, write the terms of the second factor, x and -1. Note: It does not matter which factor you put in the row or the column.
3. Multiply the first term in the first column by the first term in the row.
4. Multiply the second term in the first column by the first term in the row.
5. Multiply the first term in the first column by the second term in the row.
4 1y x x x4
x 12x x
4x
6. Multiply the second term in the first column by the second term in the row to complete your table.
4 1y x x x4
x 12x x
We want to add the terms from the table:
4x 4
2 4 4y x x x And simplify by combining like terms:
2 3 4y x x
Using this information, look at the various methods for factoring second-degree polynomials.
• Common Factoring
Given a polynomial, finding a common factor means finding the largest value that will divide each and every term evenly (greatest common factor).
Examples:
6 6 6 1x x Since 6 is the largest value that is common to each and every term, it is the greatest common factor.
b.
Since 4x is the largest value that is common to each and every term, it is the greatest common factor.
3 2 24 8 16 4 2 4x x x x x x
You can think of factoring as “undistributing”(as you saw in unit 4).
2 7 10y x x Examples:
a. Factor:
1 5 1 2y x x
5 2y x x Check using FOIL
Suppose we take this example and use the multiplication tables to help us see the process in another way. Begin by listing the factors of a that must be considered for this problem.
1 and 1List the factors of c that must be considered for this problem.
10 and 1 5 and 2
Based on the factors and all the combinations, fill in the cells in the table to see what works.
2 7 10y x x
10
2 7 10y x x
Examples:b. Factor:
3 1 2 3y x x
Check using FOIL
26 7 3y x x
6 and 1 2 and 3
3 and -1-3 and 1
Based on the factors that we will consider and all the combinations, you can fill in the following cells in the multiplication tables and test which set of factors works to give .26 7 3y x x
6x
1x
3
126x
3x
6x
326 3 3x x
6x
1x
3
126x
3x
6x
326 3 3x x
2x
3x
3
126x
9x
2x
326 7 3x x
2x
3x
3
126x
9x
2x
326 7 3x x
26 7 3y x x
26 7 3 3 1 2 3x x x x
By multiplying you see that the sum of the products does not give the middle term. In this first attempt, b = 3.
Checking: a = 6, c = -3 and b = 3
Checking: a = 6, c = -3 and b = ________
Checking: a = 6, c = -3 and b = ________
Checking: a = 6, c = -3 and b = ________
While grids offer one way to test for the correct factors, the distributive law can also be used. You will want to determine if the sum of the products is equal to b. 2 26 3 1 6 6 3 3 6 3 3x x x x x x x
2 23 6 1 6 18 3 6 17 3x x x x x x x 17
2 26 3 1 6 6 3 3 6 3 3x x x x x x x 3
2 26 1 3 6 18 3 6 17 3x x x x x x x 17
26 7 3x x
2 22 3 3 1 6 2 9 3 6 7 3x x x x x x x
While grids offer one way to test for the correct factors, the distributive law can also be used. You will want to determine if the sum of the products is equal to b. 2 26 3 1 6 6 3 3 6 3 3x x x x x x x
Checking: a = 6, c = -3 and b = ________
Since the condition for finding b has been met, the polynomial can be written in factored form as
7
26 7 3 2 3 3 1x x x x
Checking: a = 6, c = -3 and b = ________3
26 7 3x x
c. Factor:
List all the factors of a. In this example a = 1
List all factors of c. In this example c = -16 and the factors of -16 are
2 and -8 -16 and 1
-2 and 8 16 and –1 4 and - 4Test the pairs of factors of a and c to determine if the sum of their products is equal to b. In this example b = 0.
2 16x
1 and 1
1 4 1 4x x
2 16 4 4x x x
8. Change the equation of the function written in standard form to factored form. If the form cannot be changed to factored form, simply write NF to mean “not factorable”.
2a. 2x x y 2 1y x x 2b. 5 6y x x 2 3y x x
2c. 4 4x x y 2 2y x x 2d. 7 5 6f x x x NF2e. 12x x y 12x x y
2f. 3 7 20x x f x 3 5 4x x f x 2g. 10 5f x x x NF
2h. 5 125x y 5 5 5x x y 2i. 3 5 6x x y NF
2j. 4 36y x 2 6 2 6y x x 2k. 5x x y 5x x y
2l. 10 3x x f x 5 3 2 1x x f x 2m. 3 21 30f x x x 3 5 2f x x x
9. Choose any six of the factorable equations from problem 8. Graph the equations and indicate the x-intercepts. Write the intercepts as an ordered pair of the form (x,0).
2 2y x x 2 4y x
9. Choose any six of the factorable equations from problem 8. Graph the equations and indicate the x-intercepts. Write the intercepts as an ordered pair of the form (x,0).
5 1y x x 2 6 5y x x
9. Choose any six of the factorable equations from problem 8. Graph the equations and indicate the x-intercepts. Write the intercepts as an ordered pair of the form (x,0).
2 2y x x 2 4 4y x x
9. Choose any six of the factorable equations from problem 8. Graph the equations and indicate the x-intercepts. Write the intercepts as an ordered pair of the form (x,0).
12x x y 2 12y x x
9. Choose any six of the factorable equations from problem 8. Graph the equations and indicate the x-intercepts. Write the intercepts as an ordered pair of the form (x,0).
3 5 4x x f x 23 7 20f x x x
9. Choose any six of the factorable equations from problem 8. Graph the equations and indicate the x-intercepts. Write the intercepts as an ordered pair of the form (x,0).
5 5 5x x y 25 125y x
10. Examine the intercept points from the graph and the equations in factored form. What do you observe? What conclusion can you draw?
The -intercepts are the zeros of the equation.x
When finding the zeros or x-intercepts of a quadratic equation algebraically, the zeros represent the point where y = 0. To solve for x, you must set y equal to zero. Forexample, given
set y = 0 and solve by factoring.Since the product of (x + 3) and (x + 1) is zero, then one or both of these must be zero, so x = -3 and x = -1.
This means the x-intercepts are (-3,0) and (-1,0).
2 4 3y x x 20 4 3x x
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