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Algebra and Trigonometry

1.5 Factoring Polynomials

Algebra and Trigonometry1.5 Factoring Polynomials
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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. 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
    13. Chapter 13
  17. Index

Learning Objectives

In this section students will:
  • Factor the greatest common factor of a polynomial.
  • Factor a trinomial.
  • Factor by grouping.
  • Factor a perfect square trinomial.
  • Factor a difference of squares.
  • Factor the sum and difference of cubes.
  • Factor expressions using fractional or negative exponents.

Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in Figure 1.

A large rectangle with smaller squares and a rectangle inside. The length of the outer rectangle is 6x and the width is 10x. The side length of the squares is 4 and the height of the width of the inner rectangle is 4.
Figure 1

The area of the entire region can be found using the formula for the area of a rectangle.

A = lw = 10x6x = 60 x 2  units 2 A = lw = 10x6x = 60 x 2  units 2

The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of A= s 2 = 4 2 =16 A= s 2 = 4 2 =16 units2. The other rectangular region has one side of length 10x8 10x8 and one side of length 4, 4, giving an area of A=lw=4(10x8)=40x32 A=lw=4(10x8)=40x32 units2. So the region that must be subtracted has an area of 2(16)+40x32=40x 2(16)+40x32=40x units2.

The area of the region that requires grass seed is found by subtracting 60 x 2 40x 60 x 2 40x units2. This area can also be expressed in factored form as 20x(3x2) 20x(3x2) units2. We can confirm that this is an equivalent expression by multiplying.

Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

Factoring the Greatest Common Factor of a Polynomial

When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, 4 4 is the GCF of 16 16 and 20 20 because it is the largest number that divides evenly into both 16 16 and 20 20 The GCF of polynomials works the same way: 4x 4x is the GCF of 16x 16x and 20 x 2 20 x 2 because it is the largest polynomial that divides evenly into both 16x 16x and 20 x 2 . 20 x 2 .

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

Greatest Common Factor

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

How To

Given a polynomial expression, factor out the greatest common factor.

  1. Identify the GCF of the coefficients.
  2. Identify the GCF of the variables.
  3. Combine to find the GCF of the expression.
  4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.
  5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

Example 1

Factoring the Greatest Common Factor

Factor 6 x 3 y 3 +45 x 2 y 2 +21xy. 6 x 3 y 3 +45 x 2 y 2 +21xy.

Analysis

After factoring, we can check our work by multiplying. Use the distributive property to confirm that (3xy)(2 x 2 y 2 +15xy+7)=6 x 3 y 3 +45 x 2 y 2 +21xy. (3xy)(2 x 2 y 2 +15xy+7)=6 x 3 y 3 +45 x 2 y 2 +21xy.

Try It #1

Factor x( b 2 a)+6( b 2 a) x( b 2 a)+6( b 2 a) by pulling out the GCF.

Factoring a Trinomial with Leading Coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial x 2 +5x+6 x 2 +5x+6 has a GCF of 1, but it can be written as the product of the factors (x+2) (x+2) and (x+3). (x+3).

Trinomials of the form x 2 +bx+c x 2 +bx+c can be factored by finding two numbers with a product of c c and a sum of b. b. The trinomial x 2 +10x+16, x 2 +10x+16, for example, can be factored using the numbers 2 2 and 8 8 because the product of those numbers is 16 16 and their sum is 10. 10. The trinomial can be rewritten as the product of (x+2) (x+2) and (x+8). (x+8).

Factoring a Trinomial with Leading Coefficient 1

A trinomial of the form x 2 +bx+c x 2 +bx+c can be written in factored form as (x+p)(x+q) (x+p)(x+q) where pq=c pq=c and p+q=b. p+q=b.

Q&A

Can every trinomial be factored as a product of binomials?

No. Some polynomials cannot be factored. These polynomials are said to be prime.

How To

Given a trinomial in the form x 2 +bx+c, x 2 +bx+c, factor it.

  1. List factors of c. c.
  2. Find p p and q, q, a pair of factors of c c with a sum of b. b.
  3. Write the factored expression (x+p)(x+q). (x+p)(x+q).

Example 2

Factoring a Trinomial with Leading Coefficient 1

Factor x 2 +2x15. x 2 +2x15.

Analysis

We can check our work by multiplying. Use FOIL to confirm that (x3)(x+5)= x 2 +2x15. (x3)(x+5)= x 2 +2x15.

Q&A

Does the order of the factors matter?

No. Multiplication is commutative, so the order of the factors does not matter.

Try It #2

Factor x 2 7x+6. x 2 7x+6.

Factoring by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial 2 x 2 +5x+3 2 x 2 +5x+3 can be rewritten as (2x+3)(x+1) (2x+3)(x+1) using this process. We begin by rewriting the original expression as 2 x 2 +2x+3x+3 2 x 2 +2x+3x+3 and then factor each portion of the expression to obtain 2x(x+1)+3(x+1). 2x(x+1)+3(x+1). We then pull out the GCF of (x+1) (x+1) to find the factored expression.

Factor by Grouping

To factor a trinomial in the form a x 2 +bx+c a x 2 +bx+c by grouping, we find two numbers with a product of ac ac and a sum of b. b. We use these numbers to divide the x x term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.

How To

Given a trinomial in the form a x 2 +bx+c, a x 2 +bx+c, factor by grouping.
  1. List factors of ac. ac.
  2. Find p p and q, q, a pair of factors of ac ac with a sum of b. b.
  3. Rewrite the original expression as a x 2 +px+qx+c. a x 2 +px+qx+c.
  4. Pull out the GCF of a x 2 +px. a x 2 +px.
  5. Pull out the GCF of qx+c. qx+c.
  6. Factor out the GCF of the expression.

Example 3

Factoring a Trinomial by Grouping

Factor 5 x 2 +7x6 5 x 2 +7x6 by grouping.

Analysis

We can check our work by multiplying. Use FOIL to confirm that (5x3)(x+2)=5 x 2 +7x6. (5x3)(x+2)=5 x 2 +7x6.

Try It #3

Factor a. 2 x 2 +9x+9 2 x 2 +9x+9 b. 6 x 2 +x1 6 x 2 +x1

Factoring a Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

a 2 +2ab+ b 2 = (a+b) 2 and a 2 2ab+ b 2 = (ab) 2 a 2 +2ab+ b 2 = (a+b) 2 and a 2 2ab+ b 2 = (ab) 2

We can use this equation to factor any perfect square trinomial.

Perfect Square Trinomials

A perfect square trinomial can be written as the square of a binomial:

a 2 +2ab+ b 2 = (a+b) 2 a 2 +2ab+ b 2 = (a+b) 2

How To

Given a perfect square trinomial, factor it into the square of a binomial.

  1. Confirm that the first and last term are perfect squares.
  2. Confirm that the middle term is twice the product of ab. ab.
  3. Write the factored form as (a+b) 2 . (a+b) 2 .

Example 4

Factoring a Perfect Square Trinomial

Factor 25 x 2 +20x+4. 25 x 2 +20x+4.

Try It #4

Factor 49 x 2 14x+1. 49 x 2 14x+1.

Factoring a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

a 2 b 2 =(a+b)(ab) a 2 b 2 =(a+b)(ab)

We can use this equation to factor any differences of squares.

Differences of Squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

a 2 b 2 =(a+b)(ab) a 2 b 2 =(a+b)(ab)

How To

Given a difference of squares, factor it into binomials.

  1. Confirm that the first and last term are perfect squares.
  2. Write the factored form as (a+b)(ab). (a+b)(ab).

Example 5

Factoring a Difference of Squares

Factor 9 x 2 25. 9 x 2 25.

Try It #5

Factor 81 y 2 100. 81 y 2 100.

Q&A

Is there a formula to factor the sum of squares?

No. A sum of squares cannot be factored.

Factoring the Sum and Difference of Cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

a 3 + b 3 =(a+b)( a 2 ab+ b 2 ) a 3 + b 3 =(a+b)( a 2 ab+ b 2 )

Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.

a 3 b 3 =(ab)( a 2 +ab+ b 2 ) a 3 b 3 =(ab)( a 2 +ab+ b 2 )

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: Same Opposite Always Positive. For example, consider the following example.

x 3 2 3 =(x2)( x 2 +2x+4 ) x 3 2 3 =(x2)( x 2 +2x+4 )

The sign of the first 2 is the same as the sign between x 3 2 3 . x 3 2 3 . The sign of the 2x 2x term is opposite the sign between x 3 2 3 . x 3 2 3 . And the sign of the last term, 4, is always positive.

Sum and Difference of Cubes

We can factor the sum of two cubes as

a 3 + b 3 =(a+b)( a 2 ab+ b 2 ) a 3 + b 3 =(a+b)( a 2 ab+ b 2 )

We can factor the difference of two cubes as

a 3 b 3 =(ab)( a 2 +ab+ b 2 ) a 3 b 3 =(ab)( a 2 +ab+ b 2 )

How To

Given a sum of cubes or difference of cubes, factor it.

  1. Confirm that the first and last term are cubes, a 3 + b 3 a 3 + b 3 or a 3 b 3 . a 3 b 3 .
  2. For a sum of cubes, write the factored form as (a+b)( a 2 ab+ b 2 ). (a+b)( a 2 ab+ b 2 ). For a difference of cubes, write the factored form as (ab)( a 2 +ab+ b 2 ). (ab)( a 2 +ab+ b 2 ).

Example 6

Factoring a Sum of Cubes

Factor x 3 +512. x 3 +512.

Analysis

After writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be factored further. However, the trinomial portion cannot be factored, so we do not need to check.

Try It #6

Factor the sum of cubes: 216 a 3 + b 3 . 216 a 3 + b 3 .

Example 7

Factoring a Difference of Cubes

Factor 8 x 3 125. 8 x 3 125.

Analysis

Just as with the sum of cubes, we will not be able to further factor the trinomial portion.

Try It #7

Factor the difference of cubes: 1,000 x 3 1. 1,000 x 3 1.

Factoring Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance, 2 x 1 4 +5 x 3 4 2 x 1 4 +5 x 3 4 can be factored by pulling out x 1 4 x 1 4 and being rewritten as x 1 4 (2+5 x 1 2 ). x 1 4 (2+5 x 1 2 ).

Example 8

Factoring an Expression with Fractional or Negative Exponents

Factor 3x (x+2) −1 3 +4 (x+2) 2 3 . 3x (x+2) −1 3 +4 (x+2) 2 3 .

Try It #8

Factor 2 (5a1) 3 4 +7a (5a1) 1 4 . 2 (5a1) 3 4 +7a (5a1) 1 4 .

Media

Access these online resources for additional instruction and practice with factoring polynomials.

1.5 Section Exercises

Verbal

1.

If the terms of a polynomial do not have a GCF, does that mean it is not factorable? Explain.

2.

A polynomial is factorable, but it is not a perfect square trinomial or a difference of two squares. Can you factor the polynomial without finding the GCF?

3.

How do you factor by grouping?

Algebraic

For the following exercises, find the greatest common factor.

4.

14x+4xy18x y 2 14x+4xy18x y 2

5.

49m b 2 35 m 2 ba+77m a 2 49m b 2 35 m 2 ba+77m a 2

6.

30 x 3 y45 x 2 y 2 +135x y 3 30 x 3 y45 x 2 y 2 +135x y 3

7.

200 p 3 m 3 30 p 2 m 3 +40 m 3 200 p 3 m 3 30 p 2 m 3 +40 m 3

8.

36 j 4 k 2 18 j 3 k 3 +54 j 2 k 4 36 j 4 k 2 18 j 3 k 3 +54 j 2 k 4

9.

6 y 4 2 y 3 +3 y 2 y 6 y 4 2 y 3 +3 y 2 y

For the following exercises, factor by grouping.

10.

6 x 2 +5x4 6 x 2 +5x4

11.

2 a 2 +9a18 2 a 2 +9a18

12.

6 c 2 +41c+63 6 c 2 +41c+63

13.

6 n 2 19n11 6 n 2 19n11

14.

20 w 2 47w+24 20 w 2 47w+24

15.

2 p 2 5p7 2 p 2 5p7

For the following exercises, factor the polynomial.

16.

7 x 2 +48x7 7 x 2 +48x7

17.

10 h 2 9h9 10 h 2 9h9

18.

2 b 2 25b247 2 b 2 25b247

19.

9 d 2 −73d+8 9 d 2 −73d+8

20.

90 v 2 −181v+90 90 v 2 −181v+90

21.

12 t 2 +t13 12 t 2 +t13

22.

2 n 2 n15 2 n 2 n15

23.

16 x 2 100 16 x 2 100

24.

25 y 2 196 25 y 2 196

25.

121 p 2 169 121 p 2 169

26.

4 m 2 9 4 m 2 9

27.

361 d 2 81 361 d 2 81

28.

324 x 2 121 324 x 2 121

29.

144 b 2 25 c 2 144 b 2 25 c 2

30.

16 a 2 8a+1 16 a 2 8a+1

31.

49 n 2 +168n+144 49 n 2 +168n+144

32.

121 x 2 88x+16 121 x 2 88x+16

33.

225 y 2 +120y+16 225 y 2 +120y+16

34.

m 2 20m+100 m 2 20m+100

35.

25 p 2 120m+144 25 p 2 120m+144

36.

36 q 2 +60q+25 36 q 2 +60q+25

For the following exercises, factor the polynomials.

37.

x 3 +216 x 3 +216

38.

27 y 3 8 27 y 3 8

39.

125 a 3 +343 125 a 3 +343

40.

b 3 8 d 3 b 3 8 d 3

41.

64 x 3 −125 64 x 3 −125

42.

729 q 3 +1331 729 q 3 +1331

43.

125 r 3 +1,728 s 3 125 r 3 +1,728 s 3

44.

4x ( x1 ) 2 3 +3 ( x1 ) 1 3 4x ( x1 ) 2 3 +3 ( x1 ) 1 3

45.

3c ( 2c+3 ) 1 4 5 ( 2c+3 ) 3 4 3c ( 2c+3 ) 1 4 5 ( 2c+3 ) 3 4

46.

3t ( 10t+3 ) 1 3 +7 ( 10t+3 ) 4 3 3t ( 10t+3 ) 1 3 +7 ( 10t+3 ) 4 3

47.

14x ( x+2 ) 2 5 +5 ( x+2 ) 3 5 14x ( x+2 ) 2 5 +5 ( x+2 ) 3 5

48.

9y (3y13) 1 5 2 (3y13) 6 5 9y (3y13) 1 5 2 (3y13) 6 5

49.

5z (2z9) 3 2 +11 (2z9) 1 2 5z (2z9) 3 2 +11 (2z9) 1 2

50.

6d ( 2d+3 ) 1 6 +5 ( 2d+3 ) 5 6 6d ( 2d+3 ) 1 6 +5 ( 2d+3 ) 5 6

Real-World Applications

For the following exercises, consider this scenario:

Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city’s parks. The park is a rectangle with an area of 98 x 2 +105x27 98 x 2 +105x27 m2, as shown in the figure below. The length and width of the park are perfect factors of the area.

A rectangle that’s textured to look like a field. The field is labeled: l times w = ninety-eight times x squared plus one hundred five times x minus twenty-seven.
51.

Factor by grouping to find the length and width of the park.

52.

A statue is to be placed in the center of the park. The area of the base of the statue is 4 x 2 +12x+9 m 2 . 4 x 2 +12x+9 m 2 .Factor the area to find the lengths of the sides of the statue.

53.

At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain is 9 x 2 25 m 2 . 9 x 2 25 m 2 .Factor the area to find the lengths of the sides of the fountain.

For the following exercise, consider the following scenario:

A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure below. The flagpole will take up a square plot with area x 2 6x+9 x 2 6x+9 yd2.

A square that’s textured to look like a field with a missing piece in the shape of a square in the center. The sides of the larger square are labeled: 100 yards. The center square is labeled: Area: x squared minus six times x plus nine.
54.

Find the length of the base of the flagpole by factoring.

Extensions

For the following exercises, factor the polynomials completely.

55.

16 x 4 200 x 2 +625 16 x 4 200 x 2 +625

56.

81 y 4 256 81 y 4 256

57.

16 z 4 2,401 a 4 16 z 4 2,401 a 4

58.

5x ( 3x+2 ) 2 4 + ( 12x+8 ) 3 2 5x ( 3x+2 ) 2 4 + ( 12x+8 ) 3 2

59.

(32 x 3 +48 x 2 162x243) −1 (32 x 3 +48 x 2 162x243) −1

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