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SECTION 4.1 Greatest Common Factor and Factoring by Grouping
Chapter 4 Factoring
Section 4.1: Greatest Common Factor and Factoring byGrouping
GCF and Grouping
GCF and Grouping
Finding the Greatest Common Factor:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 250
CHAPTER 4 Factoring
Factoring Out the Greatest Common Factor:
University of Houston Department of Mathematics251
SECTION 4.1 Greatest Common Factor and Factoring by Grouping
Example:
Solution:
Factoring by Grouping:
MATH 1300 Fundamentals of Mathematics 252
CHAPTER 4 Factoring
Additional Example 1:
Solution:
University of Houston Department of Mathematics253
SECTION 4.1 Greatest Common Factor and Factoring by Grouping
Additional Example 2:
Solution:
The GCF is the product of the factors that are shared by all three monomials.
MATH 1300 Fundamentals of Mathematics 254
CHAPTER 4 Factoring
Additional Example 3:
Solution:
University of Houston Department of Mathematics255
SECTION 4.1 Greatest Common Factor and Factoring by Grouping
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics 256
CHAPTER 4 Factoring
Additional Example 5:
Solution:
University of Houston Department of Mathematics257
Exercise Set 4.1: Greatest Common Factor and Factoring by Grouping
Find the GCF (Greatest Common Factor) of the following monomials.
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Find the GCF of the terms of the polynomial and factor it out. If the leading coefficient is negative, then factor out the negative of the GCF.
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Factor the following expressions.
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Factor by grouping.
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MATH 1300 Fundamentals of Mathematics 258
Exercise Set 4.1: Greatest Common Factor and Factoring by Grouping
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Factor by grouping. (Hint: Use groups of three.)
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Each of the following expressions contains like terms. Do not combine the like terms; instead, simply factor by grouping. (This method will be helpful in the next section when factoring trinomials.)
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University of Houston Department of Mathematics259
SECTION 4.2 Factoring Special Binomials and Trinomials
Section 4.2: Factoring Special Binomials and Trinomials
Special Factor Patterns
Special Factor Patterns
Factoring the Difference of Two Squares:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 260
CHAPTER 4 Factoring
Note:
Factoring the Difference of Two Cubes:
Example:
University of Houston Department of Mathematics261
SECTION 4.2 Factoring Special Binomials and Trinomials
Solution:
Factoring the Sum of Two Cubes:
Example:
MATH 1300 Fundamentals of Mathematics 262
CHAPTER 4 Factoring
Solution:
Factoring Perfect Square Trinomials:
Example:
University of Houston Department of Mathematics263
SECTION 4.2 Factoring Special Binomials and Trinomials
Solution:
Additional Example 1:
Solution:
MATH 1300 Fundamentals of Mathematics 264
CHAPTER 4 Factoring
Additional Example 2:
Solution:
University of Houston Department of Mathematics265
SECTION 4.2 Factoring Special Binomials and Trinomials
(c) The monomials and share a common factor of The first step in factoring the given binomial to factor out the GCF of .
Additional Example 3:
Solution:
Additional Example 4:
MATH 1300 Fundamentals of Mathematics 266
CHAPTER 4 Factoring
Solution:
University of Houston Department of Mathematics267
Exercise Set 4.2: Factoring Special Binomials and Trinomials
Multiply the following.
1. (a)
(b)
(c)
2. (a)
(b)
(c)
Answer True or False.
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Factor the following polynomials. If the polynomial can not be factored any further within the real number system, then write the original polynomial as your answer.
11. (a)(b)(c)(d)
12. (a)(b)(c)(d)
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MATH 1300 Fundamentals of Mathematics 268
Exercise Set 4.2: Factoring Special Binomials and Trinomials
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When the remainder is zero, the dividend can be written as a product of two factors (the divisor and the quotient), as shown below.
, so .
, so
In the following examples, use either long division or synthetic division to find the quotient, and then write the dividend as a product of two factors.
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Factor the following polynomials.
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University of Houston Department of Mathematics269
SECTION 4.3 Factoring Polynomials
Section 4.3: Factoring Polynomials
Techniques for Factoring Trinomials
Techniques for Factoring Trinomials
Factorability Test for Trinomials:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 270
CHAPTER 4 Factoring
Factoring Trinomials with Leading Coefficient 1:
University of Houston Department of Mathematics271
SECTION 4.3 Factoring Polynomials
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 272
CHAPTER 4 Factoring
Factoring Trinomials with Leading Coefficient Different from 1:
University of Houston Department of Mathematics273
SECTION 4.3 Factoring Polynomials
MATH 1300 Fundamentals of Mathematics 274
CHAPTER 4 Factoring
University of Houston Department of Mathematics275
SECTION 4.3 Factoring Polynomials
Example:
Solution:
Additional Example 1:
(a)(b)
MATH 1300 Fundamentals of Mathematics 276
CHAPTER 4 Factoring
Solution:
Additional Example 2:
Solution:
University of Houston Department of Mathematics277
SECTION 4.3 Factoring Polynomials
Additional Example 3:
Solution:
MATH 1300 Fundamentals of Mathematics 278
CHAPTER 4 Factoring
Additional Example 4:
Solution:
University of Houston Department of Mathematics279
SECTION 4.3 Factoring Polynomials
Additional Example 5:
Solution:
MATH 1300 Fundamentals of Mathematics 280
Exercise Set 4.3: Factoring Polynomials
At times, it can be difficult to tell whether or not a quadratic of the form can be factored into the form , where a, b, c, d, e, f, and g are integers. If is a perfect square, then the quadratic can be factored in the above manner.For each of the following problems,
(a) Compute .
(b) Use the information from part (a) to determine whether or not the quadratic can be written as factors with integer coefficients. (Do not factor; simply answer Yes or No.)
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Factor the following polynomials. If the polynomial can not be rewritten as factors with integer coefficients, then write the original polynomial as your answer.
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Factor the following. Remember to first factor out the Greatest Common Factor (GCF) of the terms of the polynomial, and to factor out a negative if the leading coefficient is negative.
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University of Houston Department of Mathematics281
Exercise Set 4.3: Factoring Polynomials
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Factor the following polynomials. (Hint: Factor first by grouping, and then continue to factor if possible.)
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MATH 1300 Fundamentals of Mathematics 282
CHAPTER 4 Factoring
Section 4.4: Using Factoring to Solve Equations
Solving Quadratic Equations by Factoring Solving Other Polynomials Equations by Factoring
Solving Quadratic Equations by Factoring
Zero-Product Property:
Example:
Solution:
University of Houston Department of Mathematics283
SECTION 4.4 Using Factoring to Solve Equations
Example:
Solution:
The x-Intercepts of the Graph of a Quadratic Function:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 284
CHAPTER 4 Factoring
Additional Example 1:
Solution:
Additional Example 2:
University of Houston Department of Mathematics285
SECTION 4.4 Using Factoring to Solve Equations
Solution:
Additional Example 3:
Solution:
MATH 1300 Fundamentals of Mathematics 286
CHAPTER 4 Factoring
Additional Example 4:
Solution:
University of Houston Department of Mathematics287
SECTION 4.4 Using Factoring to Solve Equations
Additional Example 5:
Solution:
MATH 1300 Fundamentals of Mathematics 288
CHAPTER 4 Factoring
Additional Example 6:
Solution:
University of Houston Department of Mathematics289
SECTION 4.4 Using Factoring to Solve Equations
(c) Since , the parabola opens upward.
Solving Other Polynomial Equations by Factoring
Solving Polynomial Equations by Factoring:
Example:
MATH 1300 Fundamentals of Mathematics 290
CHAPTER 4 Factoring
Solution:
Example:
Solution:
The x-Intercepts of the Graph of a Polynomial Function:
University of Houston Department of Mathematics291
SECTION 4.4 Using Factoring to Solve Equations
Example:
Solution:
MATH 1300 Fundamentals of Mathematics 292
CHAPTER 4 Factoring
Additional Example 1:
Solution:
Additional Example 2:
Solution:
Additional Example 3:
Solution:
University of Houston Department of Mathematics293
SECTION 4.4 Using Factoring to Solve Equations
MATH 1300 Fundamentals of Mathematics 294
Exercise Set 4.4: Using Factoring to Solve Equations
Solve the following equations by factoring.
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Solve the following equations by factoring. To simplify the process, remember to first factor out the Greatest Common Factor (GCF) and to factor out a negative if the leading coefficient is negative.
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Each of the quadratic functions below is written in the form . The graph of a quadratic
function is a parabola with vertex, where
and .
(a) Find the x-intercept(s) of the parabola by setting and solving for x.
(b) Write the coordinates of the x-intercept(s) found in part (a).
(c) Find the y-intercept of the parabola and write its coordinates.
(d) Give the coordinates of the vertex (h, k) of the parabola, using the formulas and
.
(e) Does the parabola open upward (with the vertex being the lowest point on the graph) or downward (with the vertex being the highest point on the graph)?
(f) Find the axis of symmetry. (Be sure to write your answer as an equation of a line.)
(g) Draw a graph of the parabola that includes the features from parts (b) through (e).
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University of Houston Department of Mathematics295
Exercise Set 4.4: Using Factoring to Solve Equations
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Find the x-intercept(s) of the following.
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For each of the following problems:(a) Model the situation by writing appropriate
equation(s).(b) Solve the equation(s) and then answer the
question posed in the problem.
51. The length of a rectangular frame is 5 cm longer than its width. If the area of the frame is 36 cm2, find the length and width of the frame.
52. The width of a rectangular garden is 8 m shorter than its length. If the area of the field is 180 m2, find the length and the width of the garden
53. The height of a triangle is 3 cm shorter than its base. If the area of the triangle is 90 cm2, find the base and height of the triangle.
54. Find x if the area of the figure below is 26cm2. (Note that the figure may not be drawn to scale.)
MATH 1300 Fundamentals of Mathematics
x cm
x cm
3 cm
8 cm
296
