Section 13-1: The Distributive Property and Common Factors
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1 Section 13-1: The Distributive Property and Common Factors Factor: 4y 18z 4y 18z 6(4y 3z) Identify the largest factor that is common to both terms. 6 Write the epression as a product by dividing each term by the common factor. Section 13-: Multiplying and Dividing Polynomials Multiply ( + 3)( ). ( + 3)( ) Distribute each term in first factor over the entire second factor. ( ) + 3 ( ) Multiply Combine like terms Learning Outcome Multiply: (5 7)( + 3) (5 7)( + 3) Use the FOIL method to multiply Combine like terms Multiply: (5 + 3)(5 3) Sum and difference of two binomials. (5 + 3)(5 3) Square the first term. Insert a minus sign. Square the last term. 5 9 Multiply: (5 ) Binomial squared. (5 ) Square the first term. Double the product of the two terms. Square the second term Multiply: (3 )( ) (3 )( ) Recognize the product as a special product. Verify the second factor is a trinomial formed from the binomial factor in the following way: Square the first term, multiply the two terms, square the last term, and use a plus sign for all three terms. This special product gives the cube of the first term minus the cube of the second term
2 of the binomial Multiply: ( + 3)( ) ( + 3)( ) Recognize the product as a special product. Verify the second factor is a trinomial formed from the binomial factor in the following way: Square the first term, multiply the two terms, square the last term, and use a minus sign for the middle term and a plus sign for the sign of the last term. This special product gives the cube of the first term plus the cube of the second term of the binomial Learning Outcome 4 Divide ) = ( + 5) = = ( + 5) = =-3-3( + 5) = Section 13-3: Factoring Special Products Factor: Difference of two perfect squares Recognize the binomial as the "difference of two perfect squares." The first term is a perfect square, the last term is a perfect square, and the two are subtracted. (4 7)(4 + 7) Factor the epression into a product of two factors. The first factor is the "difference of the square roots" and the second factor is the "sum of the square roots" of the two terms in the given epression. Learning Outcome Write a trinomial that is a perfect-square trinomial. 36 The first term is a perfect square. 49 The last term is a perfect square. (6)(7) or 84 The middle term is twice the product of the square roots of the first and last terms.
3 Factor: Use the three terms to write the trinomial. Note: if the sign of the third term is negative, the sign of the middle term may be positive or negative, depending on the product of the square roots. Perfect-square trinomial. Recognize the trinomial as a perfect-square trinomial. The first and last terms are perfect squares (both must be positive). The middle term is twice the product of the square roots of the first and last terms. If the sign of the middle term is positive the sign of the two binomial factors will be positive. If it is negative, the sign of both binomial factors will be negative. (5 )(5 ) Factor into two binomials. The first term is the square root of the first term of the trinomial and the last term is the square root of the last term of the trinomial. Since the sign of the third term of the trinomial is a "+" the signs of the two factors are alike. And because the sign of the middle term of the trinomial is a " " the signs of the binomials are both " ". Factor: Perfect-square trinomial. Recognize the perfect-square trinomial and note that all signs are positive. (3 + )(3 +) Because all signs of the trinomial are positive, the signs of the two binomials will also be positive. Write a binomial that is the difference of two perfect cubes. 7a 3 The first term has perfect cubes for all factors. 8b 3 The last term has perfect cubes for all factors. 7a 3 8b 3 The two terms are subtracted for the "difference." Write a binomial that is the sum of two perfect cubes. 64a y 3 Both terms have perfect cubes for all factors. The two terms are added for the "sum." Factor: 7a 3 8b 3 7a 3 8b 3 Use the pattern for factoring the difference of two perfect cubes. (3a b)(9a + 6ab + 4b ) Factor: 64a y 3 64a y 3 Use the pattern for factoring the sum of two perfect cubes. (4a + 5y)(16a 0ay + 5y ) Section 13-4: Factoring General Trinomials Factor: Eamine the sign of the third term. If it is positive, find two factors of the third term whose sum is the middle term. The signs of the two binomial factors will be the same.
4 (3) or 1(6) Possible factors of 6. Select 1(6) because the sum is 7, the coefficient of the middle term. ( 1)( 6) Write the two binomials with the indicated factors. Then eamine the sign of the middle term. It is positive so the signs of both factors will be positive. ( + 1)( + 6) Factor: Eamine the sign of the third term. If it is negative, find two factors of the third term whose difference is the coefficient of the middle term. The signs of the two binomial factors will be different. (3) or 1(6) Possible factors of 6. Select (3) because the difference 3 is 1, the coefficient of the middle term. ( 3)( ) Write the two binomials with the indicated factors. Then eamine the sign of the middle term. It is positive, so the sign of the larger factor will be positive. ( + 3)( ) Learning Outcome Factor: Arrange the terms and group into two groups of two terms each. ( 7) + ( ) Be careful! Do not separate a minus sign from its term. ( 7) + ( )( 7) Factor a common factor from each of the two terms. Notice the ( 7) is a factor of both terms, thus it is a common factor and can be removed from each term. ( 7)( ) The two binomial factors should be multiplied together to be sure the product is equivalent to the initial epression. Factor by grouping: Multiply the product of the coefficients of the first and last terms. (15) = 30 Factor 30 into two factors whose sum is 13 (the coefficient of the middle term). We look for the sum because the sign of the last term is positive. 1(30), (15), 3(10), 5(6) Because the sum is 13, select 3(10) Rewrite the trinomial (middle term) using the 3 and 10 as coefficients of the two terms. Both signs are negative. ( 3) + ( ) Group the terms so each grouping has two terms. ( 3) + ( ) Factor the greatest common factor from each term. ( 3) + ( 5)( 3) Factor the common binomial factor ( 3) from each term. ( 3)( 5) Multiply to check your factoring Learning Outcome 4 Factor: To factor any epression, look for common factors first. 3( 3 10) Factor the resulting trinomial using general factoring or
5 grouping. 3( 5)( + ) Check factoring by multiplying Factoring checks.
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