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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2 ā š 2 = (a + b)(a ā b)
Sum of two cubes
š3 + š 3 = (a + b)( š2 ā ab + š 2)
Difference of two cubes
Techniques for Factoring
Polynomials
š3 ā š 3 = (a ā b)( š2 + ab + š 2)
For three terms, check for:
Perfect square trinomials
š2 + 2ab + š 2 = (š + š)2
š2 ā 2ab + š 2 = (š ā š)2
General trinomials
šš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ššš 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 3 + 250
7. 100š 8 ā 9
8. š„ 2 + x + 1
9. š 4 + š 3 ā š 2 ā c
Chapter 5
29
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
Example 1: Solve šš ā šššš + 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 + āš ā 15 = 0.
2x + āš„ ā 15 = 0
2
Original equation
2(āš„) + āš„ ā 15 = 0
Write the expression on the left in quadratic form.
(2āš„ ā5)( āš„ + 3) = 0
Factor.
2āš„ ā 5 = 0 or āš„ + 3 = 0
Zero Product Property
5
āš„ = 2 or
āš„ = ā3
Simplify.
Since the principal square root of a number cannot be negative, āš„ = 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 6 ā 16š 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 10āš„ + 21 = 0
12. š„ 3 ā 5š„ 3 + 6 = 0
10. x ā 5āš„ + 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2 + 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 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6 + š3
18. š„ 8 + 4š„ 4 + 9
19. 12š„ 4 ā 7š„ 2
20. 6š5 + 3š3 ā 1
Solve each equation.
21. š3 ā 9š2 + 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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