TERMINOLOGY 4.1. READING ASSIGNMENT 4.2 Sections 5.4, 6.1 through 6.5. Binomial. Factor (verb) GCF. Monomial. Polynomial.

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1 Section 4. Factoring Polynomials TERMINOLOGY 4.1 Prerequisite Terms: Binomial Factor (verb) GCF Monomial Polynomial Trinomial READING ASSIGNMENT 4. Sections 5.4, 6.1 through

2 READING AND SELF-DISCOVERY QUESTIONS What are the five different types of polynomials that you will learn to factor in this chapter?. What is the first thing to look for in factoring a trinomial? 3. Why is the GCF important? KEY CONCEPTS 4. The product of two binomials can be seen as a trinomial with coefficients related to the coefficients of each of the binomials. A polynomial is completely factored when none of the factors can be factored further. Conventionally, when we say factor we mean factor completely. TECHNIQUE 4. FACTORING A TRINOMIAL WHOSE COEFFICIENTS HAVE NO COMMON FACTOR You can create a table to factor a trinomial. Limitation/Caution: Most trinomials cannot be factored. For the trinomial ax + bx + c with a > 0, you can complete the table shown at right. (m) (n) (a) (c) Example: Factor 6x 9x + 8 Step 1: Replace (a) and (c) in the table. Step : Select (m) and (n) so that their sum is b and their product is the product of (a) times (c). (m) + (n) = ( 9) and (m) (n) = (a) (c) = 168 (m) (n) (m) = 8 (n) = 1 (a) = 6 (c) =

3 Step 3: Complete the table by solving the puzzle of finding the greatest common factor for each element. Place the negative signs, if any, in the column labeled (c) so that the products right to left are correct and the products down to up are correct as shown Step 4: To find the factors of the trinomial, reading left to right, a) use the down diagonal numbers as coefficients of the first binomial b) use the up diagonal numbers as the coefficients of the second binomial. = (x 7)(3x 4) 16 Why This Works To see why this method of factoring a trinomial works, multiply two binomials together using subscripted constants z 1, z, z 3, and z 4 for the coefficients of the variable x. ( z x + z )( z x + z ) = ( z z ) x + ( z z ) x + ( z z ) x + ( z z ) ( a) x + ( m) x + ( n) x + ( c) ( ) ( a) x + ( m) + ( n) x + ( c) Notice that ( a) and ( c) can be written as products of z1, z, z3, and z4. ( a) = z1z3 and ( c) = zz4 and ( a)( c) = z1z3 zz4 Also,( m) = z z and ( n) = z z and ( a)( c) = ( m)( n) = z z z z In building the table at right, the sum (m) + (n) is the coefficient of the x term and the product (m)(n) is (a)(c). Notice that the GCF of (a) and (m) is z 3. The entries in the other three cells are GCFs as well. Using the down diagonal, second factor of the trinomial in factored form is (z 3 x + z 4 ). Using the down diagonal, the first factor is (z 1 x + z ). Table of Possibly Factorable Polynomials Type Example Factors Difference of Two Squares ( terms) a b (a + b)(a b) Sum of Cubes ( terms) a 3 + b 3 (a + b)(a ab + b ) Difference of Cubes ( terms) a 3 b 3 (a b)(a ab + b ) Quadratic Trinomial (3 terms) a 5a 6 (a 6)(a + 1) Quadrinomial (4 terms) a + 3a a 6 a + 3a a 6 = (a + 3a) (a + 6) = a(a + 3) (a + 3) = (a + 3)(a ) (a) (c) (m) z 3 z (n) z 1 z 4

4 METHODOLOGY 4. FACTORING A TRINOMIAL Factoring a trinomial can assist you in solving real world problems and in graphing equations. Limitation/Caution: Many trinomials cannot be factored using just rational or real numbers. Example Factor 1x x 56x Factor 4y 3 6y + 45y Steps Discussion 1 Positive lead coefficient If necessary, factor out 1 to create a trinomial with a positive lead coefficient 3 1(1 x 58x 56) Example GCF Factor out the monomial GCF. Factor out x and simplify: x(6x 9x 8) Example 3 Trinomial Format the non-monomial factor into standard trinomial format Already done. Example 4 Factor the trinomial Use the Technique for Factoring a Trinomial. (x 7)(3x 4) 163

5 Steps Discussion Example 5 Create the factored form Present the factored form as the product of a monomial (the GCF with the correct sign) and the factors of the trinomial. x(x 7)(3x 4) Example 6 Validate Multiply the factors to verify that answer is correct x(6x 1x 8x 8) x(6x 9x 8) 3 1x + 58x 56x Example MODEL 4. Factor 3ay 4ay 7a Step 1 Positive lead coefficient Already done. Step GCF Factor out a and simplify: a ( 3y 4y 7) Step 3 Trinomial Already done. Step 4 Factor the trinomial ( y + 1)(3 y 7) Step 5 Create factored form a( y + 1)(3 y 7) Step 6 Validate a( y + 1)(3y 7) = a(3y 4y 7) = 3ay 4ay 7 a 164

6 TECHNIQUE 4. FACTORING POLYNOMIALS OF HIGHER DEGREE Step 1: Factor out the GCF Step : Try to factor as a polynomial in quadratic form Step 3: Try to factor by grouping : Factor 1a 4 10a Notice that only a 4 and a appear in the trinomial. Also, a 4 = (a ). Thus, this trinomial is in quadratic form. Thus, a 4 a ( a ) ( a ) 1 10 = a 10a 4 (6a 5a 1) a 10a = (6a + 1)( a 1) = (6a + 1)( a + 1)( a 1) Example : Factor as completely as possible: 3y + ay 9ay 6a Group the first two terms and the second two terms using parentheses, being careful to change the signs as needed. Then apply the Distributive 3y + ay 9ay 6a Property twice. ( 3y + ay) ( 9ay + 6a ) ( 3 + ) 3 ( 3 + ) y y a a y a (3y + a)( y a) CRITICAL THINKING QUESTIONS What is the standard form for trinomials?. What are m and n? 3. Why do we choose m and n such that, when multiplied, their product is equal to a c? 165

7 4. Why do we choose m and n so that m + n = b (the middle term of the quadratic expression)? 5. How do you ensure that the signs in the factored form are correct? 6. How do we validate that the factors are correct? DEMONSTRATE YOUR UNDERSTANDING 4. Factor the following. 1. x 4x 6. 3xy 15xy a 10a

8 4. + 3x 6x x ax x a 6. x 3x 4 4 IDENTIFY AND CORRECT THE ERRORS 4. In the second column, identify the error you find in each of the following worked solutions and describe the error made. Solve the problem correctly in the third column. Problem Describe Error Correct Process 1. Factor: x 5x 6 Incorrect signs in the binomial factors. x 5x 6 = ( x 3)( x ). Factor: 3b 6b 4 Failed to present the 3 factor in the factored form of the polynomial. 3b 6b 4 = + 3( b b 8) = ( b )( b + 4) 167

9 Problem Describe Error Correct Process 3. Factor: 5y 5z 3 3 Incorrectly created and used a distributive law for exponents. 5y 5z 3 3 = y z 3 3 5( ) = 5( y z) 3 4. Factor: ax 3ax + 10a Does not factor. Switched signs in the binomial factors. ax 3ax + 10a = a x + ( 3x 10) = a( x 5)( x + ) 5. Factor: y 4y 5 4 Partial factoring. y 4y 5 4 ( y ) = y 4 5 = y y + y ( )( ) 5 168