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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-5 Study Guide and Intervention
Solving Polynomial Equations
Factor Polynomials
For any number of terms, check for:
greatest common factor
For two terms, check for:
Difference of two squares
š‘Ž2 ā€“ š‘ 2 = (a + b)(a ā€“ b)
Sum of two cubes
š‘Ž3 + š‘ 3 = (a + b)( š‘Ž2 ā€“ ab + š‘ 2)
Difference of two cubes
Techniques for Factoring
Polynomials
š‘Ž3 ā€“ š‘ 3 = (a ā€“ b)( š‘Ž2 + ab + š‘ 2)
For three terms, check for:
Perfect square trinomials
š‘Ž2 + 2ab + š‘ 2 = (š‘Ž + š‘)2
š‘Ž2 ā€“ 2ab + š‘ 2 = (š‘Ž āˆ’ š‘)2
General trinomials
š‘Žš‘š‘„ 2 + (ad + bc)x + bd = (ax + b)(cx + d)
For four or more terms, check for:
Grouping
ax + bx + ay + by = x(a + b) + y(a + b)
= (a + b)(x + y)
Example: Factor šŸšŸ’š’™šŸ ā€“ 42x ā€“ 45.
First factor out the GCF to get 24š‘„ 2 ā€“ 42x ā€“ 45 = 3(8š‘„ 2 ā€“ 14x ā€“ 15). To find the coefficients of the x terms, you must find
two numbers whose product is 8 ā‹… (ā€“15) = ā€“120 and whose sum is ā€“14. The two coefficients must be ā€“20 and 6. Rewrite
the expression using ā€“20x and 6x and factor by grouping.
8š‘„ 2 ā€“ 14x ā€“ 15 = 8š‘„ 2 ā€“ 20x + 6x ā€“ 15
Group to find a GCF.
= 4x(2x ā€“ 5) + 3(2x ā€“ 5)
Factor the GCF of each binomial.
= (4x + 3 )(2x ā€“ 5)
Distributive Property
2
Thus, 24š‘„ ā€“ 42x ā€“ 45 = 3(4x + 3)(2x ā€“ 5).
Exercises
Factor completely. If the polynomial is not factorable, write prime.
1. 14š‘„ 2 š‘¦ 2 + 42xš‘¦ 3
2. 6mn + 18m ā€“ n ā€“ 3
3. 2š‘„ 2 + 18x + 16
4. š‘„ 4 ā€“ 1
5. 35š‘„ 3 š‘¦ 4 ā€“ 60š‘„ 4 y
6. 2š‘Ÿ 3 + 250
7. 100š‘š 8 ā€“ 9
8. š‘„ 2 + x + 1
9. š‘ 4 + š‘ 3 ā€“ š‘ 2 ā€“ c
Chapter 5
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Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-5 Study Guide and Intervention (continued)
Solving Polynomial Equations
Solve Polynomial Equations If a polynomial expression can be written in quadratic form, then you can use what you
know about solving quadratic equations to solve the related polynomial equation.
Example 1: Solve š’™šŸ’ ā€“ šŸ’šŸŽš’™šŸ + 144 = 0.
š‘„ 4 ā€“ 40š‘„ 2 + 144 = 0
2 2
Original equation
2
(š‘„ ) ā€“ 40(š‘„ ) + 144 = 0
2
Write the expression on the left in quadratic form.
2
(š‘„ ā€“ 4)( š‘„ ā€“ 36) = 0
š‘„2 ā€“ 4 = 0
(x ā€“ 2)(x + 2) = 0
Factor.
or
or
x ā€“ 2 = 0 or x + 2 = 0
x = 2 or
x = ā€“2
š‘„ 2 ā€“ 36 = 0
(x ā€“ 6)(x + 6) = 0
or x ā€“ 6 = 0 or x + 6 = 0
or
x = 6 or
x = ā€“6
Zero Product Property
Factor.
Zero Product Property
Simplify.
The solutions are ±2 and ±6.
Example 2: Solve 2x + āˆšš’™ ā€“ 15 = 0.
2x + āˆšš‘„ ā€“ 15 = 0
2
Original equation
2(āˆšš‘„) + āˆšš‘„ ā€“ 15 = 0
Write the expression on the left in quadratic form.
(2āˆšš‘„ ā€“5)( āˆšš‘„ + 3) = 0
Factor.
2āˆšš‘„ ā€“ 5 = 0 or āˆšš‘„ + 3 = 0
Zero Product Property
5
āˆšš‘„ = 2 or
āˆšš‘„ = ā€“3
Simplify.
Since the principal square root of a number cannot be negative, āˆšš‘„ = ā€“3 has no solution. The solution is
25
4
1
or 6 4.
Exercises
Solve each equation.
1. š‘„ 4 = 49
2. š‘„ 4 ā€“ 6š‘„ 2 = ā€“8
3. š‘„ 4 ā€“ 3š‘„ 2 = 54
4. 3š‘” 6 ā€“ 48š‘” 2 = 0
5. š‘š 6 ā€“ 16š‘š 3 + 64 = 0
6. š‘¦ 4 ā€“ 5š‘¦ 2 + 4 = 0
7. š‘„ 4 ā€“ 29š‘„ 2 + 100 = 0
8. 4š‘„ 4 ā€“ 73š‘„ 2 + 144 = 0
9.
11. x ā€“ 10āˆšš‘„ + 21 = 0
12. š‘„ 3 ā€“ 5š‘„ 3 + 6 = 0
10. x ā€“ 5āˆšš‘„ + 6 = 0
Chapter 5
30
1
š‘„2
7
ā€“ + 12= 0
š‘„
2
1
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-5 Skills Practice
Solving Polynomial Equations
Factor completely. If the polynomial is not factorable, write prime.
1. 7š‘„ 2 ā€“ 14x
2. 19š‘„ 3 ā€“ 38š‘„ 2
3. 21š‘„ 3 ā€“ 18š‘„ 2 y + 24xš‘¦ 2
4. 8š‘— 3 k ā€“ 4jš‘˜ 3 ā€“ 7
5. š‘Ž2 + 7a ā€“ 18
6. 2ak ā€“ 6a + k ā€“ 3
7. š‘2 + 8b + 7
8. š‘§ 2 ā€“ 8z ā€“ 10
9. 4š‘“ 2 ā€“ 64
10. š‘‘2 ā€“ 12d + 36
11. 9š‘„ 2 + 25
12. š‘¦ 2 + 18y + 81
13. š‘›3 ā€“ 125
14. š‘š 4 ā€“ 1
Write each expression in quadratic form, if possible.
15. 5š‘„ 4 + 2š‘„ 2 ā€“ 8
Chapter 5
16. 3š‘¦ 8 ā€“ 4š‘¦ 2 + 3
30
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
17. 100š‘Ž6 + š‘Ž3
18. š‘„ 8 + 4š‘„ 4 + 9
19. 12š‘„ 4 ā€“ 7š‘„ 2
20. 6š‘5 + 3š‘3 ā€“ 1
Solve each equation.
21. š‘Ž3 ā€“ 9š‘Ž2 + 14a = 0
22. š‘„ 3 = 3š‘„ 2
23. š‘” 4 ā€“ 3š‘” 3 ā€“ 40š‘” 2 = 0
24. š‘3 ā€“ 8š‘2 + 16b = 0
Chapter 5
30
Glencoe Algebra 2
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