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Intermediate Algebra 2e

6.2 Factor Trinomials

Intermediate Algebra 2e6.2 Factor Trinomials
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Factor trinomials of the form x2+bx+cx2+bx+c
  • Factor trinomials of the form ax2+bx+cax2+bx+c using trial and error
  • Factor trinomials of the form ax2+bx+cax2+bx+c using the ‘ac’ method
  • Factor using substitution
Be Prepared 6.4

Before you get started, take this readiness quiz.

Find all the factors of 72.
If you missed this problem, review Example 1.2.

Be Prepared 6.5

Find the product: (3y+4)(2y+5).(3y+4)(2y+5).
If you missed this problem, review Example 5.28.

Be Prepared 6.6

Simplify: −9(6);−9(6); −9(−6).−9(−6).
If you missed this problem, review Example 1.18.

Factor Trinomials of the Form x2+bx+cx2+bx+c

You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication. To factor the trinomial means to start with the product, and end with the factors.

Figure shows the equation open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses equals x squared plus 5 x plus 6. The left side of the equation is labeled factors and the right is labeled product. An arrow pointing right is labeled multiply. An arrow pointing left is labeled factor.

To figure out how we would factor a trinomial of the form x2+bx+c,x2+bx+c, such as x2+5x+6x2+5x+6 and factor it to (x+2)(x+3),(x+2)(x+3), let’s start with two general binomials of the form (x+m)(x+m) and (x+n).(x+n).

.
Foil to find the product. .
Factor the GCF from the middle terms. .
Our trinomial is of the form x2+bx+c.x2+bx+c. .

This tells us that to factor a trinomial of the form x2+bx+c,x2+bx+c, we need two factors (x+m)(x+m) and (x+n)(x+n) where the two numbers m and n multiply to c and add to b.

Example 6.9 How to Factor a Trinomial of the form x2+bx+cx2+bx+c

Factor: x2+11x+24.x2+11x+24.

Try It 6.17

Factor: q2+10q+24.q2+10q+24.

Try It 6.18

Factor: t2+14t+24.t2+14t+24.

Let’s summarize the steps we used to find the factors.

How To

Factor trinomials of the form x2+bx+c.x2+bx+c.

  1. Step 1. Write the factors as two binomials with first terms x. x2+bx+c(x)(x)x2+bx+c(x)(x)
  2. Step 2. Find two numbers m and n that
    • multiply to c,m·n=cc,m·n=c
    • add to b,m+n=bb,m+n=b
  3. Step 3. Use m and n as the last terms of the factors. (x+m)(x+n)(x+m)(x+n)
  4. Step 4. Check by multiplying the factors.

In the first example, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let’s look first at trinomials with only the middle term negative.

How do you get a positive product and a negative sum? We use two negative numbers.

Example 6.10

Factor: y211y+28.y211y+28.

Try It 6.19

Factor: u29u+18.u29u+18.

Try It 6.20

Factor: y216y+63.y216y+63.

Now, what if the last term in the trinomial is negative? Think about FOIL. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.

How do you get a negative product and a positive sum? We use one positive and one negative number.

When we factor trinomials, we must have the terms written in descending order—in order from highest degree to lowest degree.

Example 6.11

Factor: 2x+x248.2x+x248.

Try It 6.21

Factor: 9m+m2+18.9m+m2+18.

Try It 6.22

Factor: −7n+12+n2.−7n+12+n2.

Sometimes you’ll need to factor trinomials of the form x2+bxy+cy2x2+bxy+cy2 with two variables, such as x2+12xy+36y2.x2+12xy+36y2. The first term, x2,x2, is the product of the first terms of the binomial factors, x·x.x·x. The y2y2 in the last term means that the second terms of the binomial factors must each contain y. To get the coefficients b and c, you use the same process summarized in How To Factor trinomials.

Example 6.12

Factor: r28rs9s2.r28rs9s2.

Try It 6.23

Factor: a211ab+10b2.a211ab+10b2.

Try It 6.24

Factor: m213mn+12n2.m213mn+12n2.

Some trinomials are prime. The only way to be certain a trinomial is prime is to list all the possibilities and show that none of them work.

Example 6.13

Factor: u29uv12v2.u29uv12v2.

Try It 6.25

Factor: x27xy10y2.x27xy10y2.

Try It 6.26

Factor: p2+15pq+20q2.p2+15pq+20q2.

Let’s summarize the method we just developed to factor trinomials of the form x2+bx+c.x2+bx+c.

Strategy for Factoring Trinomials of the Form x2+bx+cx2+bx+c

When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.

x2+bx+c(x+m)(x+n)Whencis positive,mandnhave the same sign.bpositivebnegativem,npositivem,nnegativex2+5x+6x26x+8(x+2)(x+3)(x4)(x2)same signssame signsWhencis negative,mandnhave opposite signs.x2+x12x22x15(x+4)(x3)(x5)(x+3)opposite signsopposite signsx2+bx+c(x+m)(x+n)Whencis positive,mandnhave the same sign.bpositivebnegativem,npositivem,nnegativex2+5x+6x26x+8(x+2)(x+3)(x4)(x2)same signssame signsWhencis negative,mandnhave opposite signs.x2+x12x22x15(x+4)(x3)(x5)(x+3)opposite signsopposite signs

Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.

Factor Trinomials of the form ax2 + bx + c using Trial and Error

Our next step is to factor trinomials whose leading coefficient is not 1, trinomials of the form ax2+bx+c.ax2+bx+c.

Remember to always check for a GCF first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods we’ve used so far. Let’s do an example to see how this works.

Example 6.14

Factor completely: 4x3+16x220x.4x3+16x220x.

Try It 6.27

Factor completely: 5x3+15x220x.5x3+15x220x.

Try It 6.28

Factor completely: 6y3+18y260y.6y3+18y260y.

What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.

Let’s factor the trinomial 3x2+5x+2.3x2+5x+2.

From our earlier work, we expect this will factor into two binomials.

3x2+5x+2()()3x2+5x+2()()

We know the first terms of the binomial factors will multiply to give us 3x2.3x2. The only factors of 3x23x2 are 1x,3x.1x,3x. We can place them in the binomials.

The polynomial is 3x squared plus 5x plus 2. There are two pairs of parentheses, with the first terms in them being x and 3x.

Check: Does 1x·3x=3x2?1x·3x=3x2?

We know the last terms of the binomials will multiply to 2. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of 2 are 1, 2. But we now have two cases to consider as it will make a difference if we write 1, 2 or 2, 1.

Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses.

Which factors are correct? To decide that, we multiply the inner and outer terms.

Figure shows the polynomial 3x squared plus 5x plus 2 and two possible pairs of factors. One is open parentheses x plus 1 close parentheses open parentheses 3x plus 2 close parentheses. The other is open parentheses x plus 2 close parentheses open parentheses 3x plus 1 close parentheses. In each case, arrows are shown pairing the first term of the first factor with the last term of the second factor and the first term of the second factor with the last term of the first factor.

Since the middle term of the trinomial is 5x,5x, the factors in the first case will work. Let’s use FOIL to check.

(x+1)(3x+2)3x2+2x+3x+23x2+5x+2(x+1)(3x+2)3x2+2x+3x+23x2+5x+2

Our result of the factoring is:

3x2+5x+2(x+1)(3x+2)3x2+5x+2(x+1)(3x+2)

Example 6.15 How to Factor a Trinomial Using Trial and Error

Factor completely using trial and error: 3y2+22y+7.3y2+22y+7.

Try It 6.29

Factor completely using trial and error: 2a2+5a+3.2a2+5a+3.

Try It 6.30

Factor completely using trial and error: 4b2+5b+1.4b2+5b+1.

How To

Factor trinomials of the form ax2+bx+cax2+bx+c using trial and error.

  1. Step 1. Write the trinomial in descending order of degrees as needed.
  2. Step 2. Factor any GCF.
  3. Step 3. Find all the factor pairs of the first term.
  4. Step 4. Find all the factor pairs of the third term.
  5. Step 5. Test all the possible combinations of the factors until the correct product is found.
  6. Step 6. Check by multiplying.

Remember, when the middle term is negative and the last term is positive, the signs in the binomials must both be negative.

Example 6.16

Factor completely using trial and error: 6b213b+5.6b213b+5.

Try It 6.31

Factor completely using trial and error: 8x214x+3.8x214x+3.

Try It 6.32

Factor completely using trial and error: 10y237y+7.10y237y+7.

When we factor an expression, we always look for a greatest common factor first. If the expression does not have a greatest common factor, there cannot be one in its factors either. This may help us eliminate some of the possible factor combinations.

Example 6.17

Factor completely using trial and error: 18x237xy+15y2.18x237xy+15y2.

Try It 6.33

Factor completely using trial and error 18x23xy10y2.18x23xy10y2.

Try It 6.34

Factor completely using trial and error: 30x253xy21y2.30x253xy21y2.

Don’t forget to look for a GCF first and remember if the leading coefficient is negative, so is the GCF.

Example 6.18

Factor completely using trial and error: −10y455y360y2.−10y455y360y2.

Try It 6.35

Factor completely using trial and error: 15n385n2+100n.15n385n2+100n.

Try It 6.36

Factor completely using trial and error: 56q3+320q296q.56q3+320q296q.

Factor Trinomials of the Form ax2+bx+cax2+bx+c using the “ac” Method

Another way to factor trinomials of the form ax2+bx+cax2+bx+c is the “ac” method. (The “ac” method is sometimes called the grouping method.) The “ac” method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!

Example 6.19 How to Factor Trinomials using the “ac” Method

Factor using the ‘ac’ method: 6x2+7x+2.6x2+7x+2.

Try It 6.37

Factor using the ‘ac’ method: 6x2+13x+2.6x2+13x+2.

Try It 6.38

Factor using the ‘ac’ method: 4y2+8y+3.4y2+8y+3.

The “ac” method is summarized here.

How To

Factor trinomials of the form ax2+bx+cax2+bx+c using the “ac” method.

  1. Step 1. Factor any GCF.
  2. Step 2. Find the product ac.
  3. Step 3. Find two numbers m and n that:
    Multiply toacm·n=a·cAdd tobm+n=bax2+bx+cMultiply toacm·n=a·cAdd tobm+n=bax2+bx+c
  4. Step 4. Split the middle term using m and n. ax2+mx+nx+cax2+mx+nx+c
  5. Step 5. Factor by grouping.
  6. Step 6. Check by multiplying the factors.

Don’t forget to look for a common factor!

Example 6.20

Factor using the ‘ac’ method: 10y255y+70.10y255y+70.

Try It 6.39

Factor using the ‘ac’ method: 16x232x+12.16x232x+12.

Try It 6.40

Factor using the ‘ac’ method: 18w239w+18.18w239w+18.

Factor Using Substitution

Sometimes a trinomial does not appear to be in the ax2+bx+cax2+bx+c form. However, we can often make a thoughtful substitution that will allow us to make it fit the ax2+bx+cax2+bx+c form. This is called factoring by substitution. It is standard to use u for the substitution.

In the ax2+bx+c,ax2+bx+c, the middle term has a variable, x, and its square, x2,x2, is the variable part of the first term. Look for this relationship as you try to find a substitution.

Example 6.21

Factor by substitution: x44x25.x44x25.

Try It 6.41

Factor by substitution: h4+4h212.h4+4h212.

Try It 6.42

Factor by substitution: y4y220.y4y220.

Sometimes the expression to be substituted is not a monomial.

Example 6.22

Factor by substitution: (x2)2+7(x2)+12(x2)2+7(x2)+12

Try It 6.43

Factor by substitution: (x5)2+6(x5)+8.(x5)2+6(x5)+8.

Try It 6.44

Factor by substitution: (y4)2+8(y4)+15.(y4)2+8(y4)+15.

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with factoring.

Section 6.2 Exercises

Practice Makes Perfect

Factor Trinomials of the Form x2+bx+cx2+bx+c

In the following exercises, factor each trinomial of the form x2+bx+c.x2+bx+c.

61.

p2+11p+30p2+11p+30

62.

w2+10x+21w2+10x+21

63.

n2+19n+48n2+19n+48

64.

b2+14b+48b2+14b+48

65.

a2+25a+100a2+25a+100

66.

u2+101u+100u2+101u+100

67.

x28x+12x28x+12

68.

q213q+36q213q+36

69.

y218y+45y218y+45

70.

m213m+30m213m+30

71.

x28x+7x28x+7

72.

y25y+6y25y+6

73.

5p6+p25p6+p2

74.

6n7+n26n7+n2

75.

86x+x286x+x2

76.

7x+x2+67x+x2+6

77.

x21211xx21211x

78.

−1110x+x2−1110x+x2

In the following exercises, factor each trinomial of the form x2+bxy+cy2.x2+bxy+cy2.

79.

x22xy80y2x22xy80y2

80.

p28pq65q2p28pq65q2

81.

m264mn65n2m264mn65n2

82.

p22pq35q2p22pq35q2

83.

a2+5ab24b2a2+5ab24b2

84.

r2+3rs28s2r2+3rs28s2

85.

x23xy14y2x23xy14y2

86.

u28uv24v2u28uv24v2

87.

m25mn+30n2m25mn+30n2

88.

c27cd+18d2c27cd+18d2

Factor Trinomials of the Form ax2+bx+cax2+bx+c Using Trial and Error

In the following exercises, factor completely using trial and error.

89.

p38p220pp38p220p

90.

q35q224qq35q224q

91.

3m321m2+30m3m321m2+30m

92.

11n355n2+44n11n355n2+44n

93.

5x4+10x375x25x4+10x375x2

94.

6y4+12y348y26y4+12y348y2

95.

2t2+7t+52t2+7t+5

96.

5y2+16y+115y2+16y+11

97.

11x2+34x+311x2+34x+3

98.

7b2+50b+77b2+50b+7

99.

4w25w+14w25w+1

100.

5x217x+65x217x+6

101.

4q27q24q27q2

102.

10y253y1110y253y11

103.

6p219pq+10q26p219pq+10q2

104.

21m229mn+10n221m229mn+10n2

105.

4a2+17ab15b24a2+17ab15b2

106.

6u2+5uv14v26u2+5uv14v2

107.

−16x232x16−16x232x16

108.

−81a2+153a+18−81a2+153a+18

109.

−30q3140q280q−30q3140q280q

110.

−5y330y2+35y−5y330y2+35y

Factor Trinomials of the Form ax2+bx+cax2+bx+c using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

111.

5n2+21n+45n2+21n+4

112.

8w2+25w+38w2+25w+3

113.

4k216k+154k216k+15

114.

5s29s+45s29s+4

115.

6y2+y156y2+y15

116.

6p2+p226p2+p22

117.

2n227n452n227n45

118.

12z241z1112z241z11

119.

60y2+290y5060y2+290y50

120.

6u246u166u246u16

121.

48z3102z245z48z3102z245z

122.

90n3+42n2216n90n3+42n2216n

123.

16s2+40s+2416s2+40s+24

124.

24p2+160p+9624p2+160p+96

125.

48y2+12y3648y2+12y36

126.

30x2+105x6030x2+105x60

Factor Using Substitution

In the following exercises, factor using substitution.

127.

x46x27x46x27

128.

x4+2x28x4+2x28

129.

x43x228x43x228

130.

x413x230x413x230

131.

(x3)25(x3)36(x3)25(x3)36

132.

(x2)23(x2)54(x2)23(x2)54

133.

(3y2)2(3y2)2(3y2)2(3y2)2

134.

(5y1)23(5y1)18(5y1)23(5y1)18

Mixed Practice

In the following exercises, factor each expression using any method.

135.

u212u+36u212u+36

136.

x214x32x214x32

137.

r220rs+64s2r220rs+64s2

138.

q229qr96r2q229qr96r2

139.

12y229y+1412y229y+14

140.

12x2+36y24z12x2+36y24z

141.

6n2+5n46n2+5n4

142.

3q2+6q+23q2+6q+2

143.

13z2+39z2613z2+39z26

144.

5r2+25r+305r2+25r+30

145.

3p2+21p3p2+21p

146.

7x221x7x221x

147.

6r2+30r+366r2+30r+36

148.

18m2+15m+318m2+15m+3

149.

24n2+20n+424n2+20n+4

150.

4a2+5a+24a2+5a+2

151.

x44x212x44x212

152.

x47x28x47x28

153.

(x+3)29(x+3)36(x+3)29(x+3)36

154.

(x+2)225(x+2)54(x+2)225(x+2)54

Writing Exercises

155.

Many trinomials of the form x2+bx+cx2+bx+c factor into the product of two binomials (x+m)(x+n).(x+m)(x+n). Explain how you find the values of m and n.

156.

Tommy factored x2x20x2x20 as (x+5)(x4).(x+5)(x4). Sara factored it as (x+4)(x5).(x+4)(x5). Ernesto factored it as (x5)(x4).(x5)(x4). Who is correct? Explain why the other two are wrong.

157.

List, in order, all the steps you take when using the “ac” method to factor a trinomial of the form ax2+bx+c.ax2+bx+c.

158.

How is the “ac” method similar to the “undo FOIL” method? How is it different?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 columns, 4 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don’t get it. The first column has the following statements: factor trinomials of the form x squared plus bx plus c, factor trinomials of the form a x squared plus b x plus c using trial and error, factor trinomials of the form a x squared plus bx plus c with using the “ac” method, factor using substitution.

After reviewing this checklist, what will you do to become confident for all objectives?

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