Digital Lessons on Factoring Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring using the Greatest Common Factor and Factoring by Grouping 1. List all possible factors for a given number. 2. Find the greatest common factor of a set of numbers or monomials. 3. Write a polynomial as a product of a monomial GCF and a polynomial. 4. Factor by grouping. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Following are some examples of factored form: Factored form: We say that a number or expression is in its factored form if it is written as a product of factors. Following are some examples of factored form: An integer written in factored form with integer factors: 28 = 2 • 14 A monomial written in factored form with monomial factors: 8x5 = 4x2 • 2x3 A polynomial written in factored form with a monomial factor and a polynomial factor: 2x + 8 = 2(x + 4) A polynomial written in factored form with two polynomial factors: x2 + 5x + 6 = (x + 2)(x + 3) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Listing all possible factors for a given number. Example: List all natural number factors of 24. Solution: To list all the natural number factors, we can divide 24 by 1, 2, 3, and so on, writing each divisor and quotient pair as a product until we have all possible combinations. 1 • 24, 2 • 12, 3 • 8, 4 • 6 The natural number factors of 24 are 1, 2, 3, 4, 6, 8, 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Find the greatest common factor of a set of numbers or monomials. Greatest common factor (GCF) is the largest natural number that divides all given numbers perfectly or without remainder. Steps for finding the GCF of a set of numbers by listing: 1. List all possible factors for each given number. 2. Search the lists for the largest factor common to all lists. Example: Find the GCF of 24 and 36. Solution: Factors of 24: 1, 2, 3, 4, 6, 8, 12 Factors of 36: 1, 2, 3, 4,6, 9, 12,18 The GCF of 24 and 36 is 12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To find the greatest common factor of a given set of numbers: Prime Factorization Method for Finding GCF To find the greatest common factor of a given set of numbers: 1. Write the prime factorization of each given number in exponential form. 2. Create a factorization for the GCF that includes only those prime factors common to all the factorizations, each raised to its smallest exponent in the factorization. 3. Multiply the factors in the factorization created in Step 2. Note: If there are no common prime factors, then the GCF is 1. Example: Find the GCF of 25a3b and 40a2. Solution: Write the prime factorization of each monomial, treating the variables like prime factors. 25a3b = 5 • 5 • a3 • b 40a2 = 2 • 2 •2 • 5 • a2 GCF = 5 •a2 = 5a2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring a Monomial GCF Out of a Polynomial To factor a monomial GCF out of a given polynomial: 1. Find the GCF of the terms that make up the polynomial. 2. Rewrite the given polynomial as a product of the GCF and parentheses that contain the result of dividing the given polynomial by the GCF. Given polynomial = GCF Example: Factor: Solution Find the GCF of Because the first term in the polynomial is negative, we will factor out the negative of the GCF to avoid a negative first term inside the parentheses. We will factor out 3x2 y. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

continued 2. Write the given polynomial as the product of the GCF and the parentheses containing the quotient of the given polynomial and the GCF. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. Solution: Notice that this expression is a sum of two products, a and (b + 5), and 8 and (b + 5). Further, note that (b + 5) is the GCF of the two products. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To factor a four-term polynomial by grouping: Factoring by grouping. To factor a four-term polynomial by grouping: 1. Factor out any monomial GCF (other than 1) that is common to all four terms. 2. Group together pairs of terms and factor the GCF out of each pair. 3. If there is a common binomial factor, then factor it out. 4. If there is no common binomial factor, then interchange the middle two terms and repeat the process. If there is still no common binomial factor, then the polynomial cannot be factored by grouping. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. Solution: First we look for a monomial GCF (other than 1). This polynomial does not have one. Because the polynomial has four terms, we now try to factor by grouping. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring Trinomials of the Form x2 + bx + c To factor a trinomial of the form x2 + bx + c : 1. Find two numbers with a product equal to c and a sum equal to b. 2. The factored trinomial will have the form: (x + first number) (x + second number). Note: The signs in the binomial factors can be minus signs, depending on the signs of b and c. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. x2 – 6x + 8 Solution: We must find a pair of numbers whose product is 8 and whose sum is –6. If two numbers have a positive product and negative sum, they must both be negative. Following is a table listing the products and sums: Product Sum (–1)(–8) = 8 –1 + (–8) = –9 (–2)(–4) = 8 –2 + (–4) = –6 This is the correct combination. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

continued Answer x2 – 6x + 8 = (x – 2)(x – 4) Check We can check by multiplying the binomial factors to see if their product is the original polynomial. (x – 2)(x – 4) = x2 – 4x – 2x + 8 Multiply the factors using FOIL. = x2 – 6x + 8 The product is the original polynomial. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. a2 – ab – 20b2 Solution: We must find a pair of terms whose product is 20b2 and whose sum is –1b. These terms would have to be –5b and 4b. Answer a2 – ab – 20b2 = (a – 5b)(a + 4b) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example: Factor. 4xy3 + 12xy2 – 72xy Factor out a monomial GCF, then factor the trinomial of the form x2 + bx + c. Example: Factor. 4xy3 + 12xy2 – 72xy Solution First, we look for a monomial GCF (other than 1). Notice that the GCF of the terms is 4xy. Factoring out the monomial, we have 4xy3 + 12xy2 – 72xy = 4xy(y2 + 3y – 18) Now try to factor the trinomial to two binomials. We must find a pair of numbers whose product is –18 and whose sum is 3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Answer 4xy3 + 12xy2 – 72xy = 4xy(y – 3)(y + 6) continued Product Sum (–1)(18) = –18 –1 + 18 = 17 (–2)(9) = – 18 –2 + 9 = 7 (–3)(6) = – 18 –3 + 6 = 3 Answer This is the correct combination. 4xy3 + 12xy2 – 72xy = 4xy(y – 3)(y + 6) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring Trinomials of the Form ax2 + bx + c, where a  1 Factoring by Trial and Error To factor a trinomial of the form ax2 + bx + c, where a 1, by trial and error: 1. Look for a monomial GCF in all the terms. If there is one, factor it out. 2. Write a pair of first terms whose product is ax2. ax2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

c Inner + Outer bx 3. Write a pair of last terms whose product is c. 4. Verify that the sum of the inner and outer products is bx (the middle term of the trinomial). c + Outer bx Inner Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

If the sum of the inner and outer products is not bx, then try the following: a. Exchange the first terms of the binomials from step 3, then repeat step 4. b. Exchange the last terms of the binomials from step 3, then repeat step 4. c. Repeat steps 2 – 4 with a different combination of first and last terms. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. Solution The first terms must multiply to equal 6x2. These could be x and 6x, or 2x and 3x. The last terms must multiply to equal –5. Because –5 is negative, the last terms in the binomials must have different signs. This factor pair must be 1 and 5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

continued Now we multiply binomials with various combinations of these first and last terms until we find a combination whose inner and outer products combine to equal 13x. Incorrect combinations. Correct combination. Answer Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Now we factor the trinomial within the parentheses. Example Factor. Solution First, we factor out the monomial GCF, 3x. Now we factor the trinomial within the parentheses. The first terms must multiply to equal 7x2. These could be x and 7x. The last terms must multiply to equal 3. Because 3 is a prime number, its factors are 1 and 3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

continued Now we multiply binomials with various combinations of these first and last terms until we find a combination whose inner and outer products combine to equal –20x. Correct combination. Answer Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring ax2 + bx + c, where a 1, by Grouping To factor a trinomial of the form ax2 + bx + c, where a 1, by grouping: 1. Look for a monomial GCF in all the terms. If there is one, factor it out. 2. Multiply a and c. 3. Find two factors of this product whose sum is b. 4. Write a four-term polynomial in which bx is written as the sum of two like terms whose coefficients are the two factors you found in step 3. 5. Factor by grouping. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. Solution Notice that for this trinomial, a = 2, b = –15, and c = 7. We begin my multiplying a and c: (2)(7) = 14. Now we find two factors of 14 whose sum is –15. Notice that these two factors must both be negative. Factors of ac Sum of Factors of ac (–2)(–7) = 14 –2 + (–7) = –9 (–1)(–14) = 14 –1 + (– 14) = –15 Correct Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

continued –15x –x – 14x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring Special Products 1. Factor perfect square trinomials. 2. Factor a difference of squares. 3. Factor a difference of cubes. 4. Factor a sum of cubes. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring Perfect Square Trinomials a2 + 2ab + b2 = (a + b)2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. 9a2 + 6a + 1 Solution This trinomial is a perfect square because the first and the last terms are perfect squares and twice the product of their roots is the middle term. 9a2 + 6a + 1 The square root of 9a2 is 3a. The square root of 1 is 1. Twice the product of 3a and 1 is (2)(3a)(1) = 6a, which is the middle term. Answer 9a2 + 6a + 1 = (3a + 1)2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. 16x2 – 56x + 49 Solution This trinomial is a perfect square. 16x2 – 56x + 49 The square root of 16x2 is 4x. The square root of 49 is 7. Twice the product of 4x and 7 is (2)(4x)(7) = 56x, which is the middle term. Answer 16x2 – 56x + 49 = (4x – 7)2 Use a2 – 2ab + b2 = (a – b)2, where a = 4x and b = 7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring a Difference of Squares a2 – b2 = (a + b)(a – b) Warning: A sum of squares a2 + b2 is prime and cannot be factored. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. 9x2 – 16y2 Solution This binomial is a difference of squares because 9x2 – 16y2 = (3x)2 – (4y)2 . To factor it, we use the rule a2 – b2 = (a + b)(a – b). a2 – b2 = (a + b)(a – b) 9x2 – 16y2 = (3x)2 – (4y)2 = (3x + 4y)(3x – 4y) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. n4 – 625 Solution This binomial is a difference of squares, where a = n2 and b = 25. n4 – 625 = (n2 + 25)(n2 – 25) Use a2 – b2 = (a + b)(a – b). = (n2 + 25)(n + 5)(n – 5) Factor n2 – 25, using a2 – b2 = (a + b)(a – b) with a = n and b = 5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring a Difference of Cubes a3 – b3 = (a – b)(a2 + ab + b2) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

216x3 – 64 = (6x)3 – (4)3 = (6x – 4)((6x)2 + (6x)(4) + (4)2) Example Factor. 216x3 – 64 Solution This binomial is a difference of cubes. a3 – b3 = (a – b) (a2 + a b + b2) 216x3 – 64 = (6x)3 – (4)3 = (6x – 4)((6x)2 + (6x)(4) + (4)2) = (6x – 4)(36x2 + 24x + 16) Note: The trinomial may seem like a perfect square. However, to be a perfect square, the middle term should be 2ab. In this trinomial, we only have ab, so it cannot be factored. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring a Sum of Cubes a3 + b3 = (a + b)(a2 – ab + b2) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

= 6x(1 + 3y)((1)2 – (1)(3y) + (3y)2) Example Factor. 6x +162xy3 Solution The terms in this binomial have a monomial GCF, 6x. 6x +162xy3 = 6x(1 + 27y3) = 6x(1 + 3y)((1)2 – (1)(3y) + (3y)2) = 6x(1 + 3y)(1 – 3y + 9y2) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Strategies for Factoring Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring a Polynomial To factor a polynomial, first factor out any monomial GCF, then consider the number of terms in the polynomial. If the polynomial has: I. Four terms, then try to factor by grouping II. Three terms, then determine if the trinomial is a perfect square or not. A. If the trinomial is a perfect square, then consider its form. 1. If in the form a2 + 2ab + b2, then the factored form is (a + b)2. 2. If in the form a2  2ab + b2, then the factored form is (a  b)2. B. If the trinomial is not a perfect square, then consider its form. 1. If in the form x2 + bx + c, then find two factors of c whose sum is b, and write the factored form as (x + first number)(x + second number). Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Note: Always look to see if any of the factors can be factored. Factoring a Polynomial continued 2. If in the form ax2 + bx + c, where a  1, then use trial and error. Or, find two factors of ac whose sum is b; write these factors as coefficients of two like terms that, when combined, equal bx; and then factor by grouping. III. Two terms, then determine if the binomial is a difference of squares, sum of cubes, or difference of cubes. A. If given a binomial that is a difference of squares, a2 – b2, then the factors are conjugates and the factored form is (a + b)(a – b). Note that a sum of squares cannot be factored. B. If given a binomial that is a sum of cubes, a3 + b3, then the factored form is (a + b)(a2 – ab + b2). C. If given a binomial that is a difference of cubes, a3 – b3, then the factored form is (a – b)(a2 + ab + b2). Note: Always look to see if any of the factors can be factored. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. 12x2 – 8x – 15 Solution There is no GCF. Not a perfect square, since the first and last terms are not perfect squares. Use trial and error or grouping. (x – 3)(12x + 5) = 12x2 + 5x – 36x – 15 No (6x – 3)(2x + 3) = 12x2 + 18x – 6x – 9 No (6x + 5)(2x – 3) = 12x2 – 18x + 10x – 15 = 12x2 – 8x – 15 Correct Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

5x(x2 – 2x – 24) Factored out the monomial GCF, 5x. Example Factor. 5x3 – 10x2 – 120x Solution 5x(x2 – 2x – 24) Factored out the monomial GCF, 5x. Look for two numbers whose product is –24 and whose sum is 2. 5x3 – 10x2 – 120x =5x(x + 4)(x – 6) Product Sum (1)(24) = 24 1 + 24 = 23 4(6) = 24 4 + (6) = 2 Correct combination. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. 8a4 – 72n2 Solution 8a4 – 72n2 = 8(a4 – 9n2) Factor out the monomial GCF, 8. a4 – 9n2 is a difference of squares = 8(a2 – 3n)(a2 + 3n) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. 12y5 + 84y3 Solution 12y3(y2 + 7) Factor out the monomial GCF, 12y3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. 150x3y – 120x2y2 + 24xy3 Solution 6xy(25x2 – 20xy + 4y2) 6xy(5x – 2y)2 Factor out the monomial GCF, 6xy. Factor the perfect square trinomial. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor. x5 – 2x3 – 27x2 + 54 Solution No common monomial, factor by grouping. (x5 – 2x3)(– 27x2 + 54) x3(x2 – 2)–27(x2 – 2) (x2 – 2)(x3 – 27) Difference of cubes (x2 – 2)(x – 3)(x2 + 3x + 9) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solving Quadratic Equations by Factoring . Zero-Factor Theorem If a and b are real numbers and ab = 0, then a = 0 or b = 0. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solve. (x + 4)(x + 5) = 0 Solution According to the zero-factor theorem, one of the two factors, or both factors, must equal 0. x + 4 = 0 or x + 5 = 0 Solve each equation. x = 4 x = 5 Check For x = 4: For x = 5: (x + 4)(x + 5) = 0 (x + 4)(x + 5) = 0 (4 + 4)(4 + 1) = 0 (5 + 4)(5 + 5) = 0 0(3) = 0 (1)(0) = 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solving Equations with Two or More Factors Equal to 0 To solve an equation in which two or more factors are equal to 0, use the zero-factor theorem: 1. Set each factor equal to zero. 2. Solve each of those equations. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To check, we verify that the solutions satisfy the original equations. Example Solve. a. y(5y + 2) = 0 b. x(x + 2)(5x – 4) = 0 Solution a. y(5y + 2) = 0 y = 0 or 5y + 2 = 0 5y = 2 b. This equation is already solved. To check, we verify that the solutions satisfy the original equations. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solve quadratic equations by factoring. Quadratic equation in one variable: An equation that can be written in the form ax2 + bx + c = 0, where a, b, and c are all real numbers and a  0. Solving Quadratic Equations Using Factoring To solve a quadratic equation: 1. Write the equation in standard form (ax2 + bx + c = 0). 2. Write the variable expression in factored form. 3. Use the zero-factor theorem to solve. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solve. 2x2 – 5x – 3 = 0 Solution The equation is in standard form, so we can factor. 2x2 – 5x – 3 = 0 (2x + 1)(x – 3) = 0 Use the zero-factor theorem to solve. 2x + 1 = 0 or x – 3 = 0 To check, we verify that the solutions satisfy the original equations. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solve. 6y2 + 11y = 10 + 4y Solution Write the equation in standard form. 6y2 + 11y = 10 + 4y 6y2 + 7y = 10 Subtract 4y from both sides. 6y2 + 7y – 10 = 0 Subtract 10 from both sides. (6y – 5)(y + 2) = 0 Factor. 6y – 5 = 0 or y + 2 = 0 Use the zero-factor theorem. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley