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Transcript
Factoring Polynomials The Greatest Common Factor Factors Factors Factors are numbers (or polynomials) you can multiply together to get another number (or polynomial). Factoring – writing a polynomial as a product of polynomials. Martin-Gay, Developmental Mathematics 3 Greatest Common Factor Prime Numbers – numbers greater than one, have only factors of 1 and itself Composite Number – numbers greater than on e and have more than 2 factors Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved. Finding the GCF of a List of Integers or Terms 1) Prime factor the numbers. 2) Identify common prime factors. 3) Take the product of all common prime factors. • If there are no common prime factors, GCF is 1. Martin-Gay, Developmental Mathematics 4 Greatest Common Factor Example Find the GCF of each list of numbers. 1) 12 and 8 12 = 2 · 2 · 3 8=2·2·2 So the GCF is 2 · 2 = 4. 2) 7 and 20 7=1·7 20 = 2 · 2 · 5 There are no common prime factors so the GCF is 1. Martin-Gay, Developmental Mathematics 5 Greatest Common Factor GCF of monomials or terms – 1. write the gcf of the coefficients. 2. write the variables they have in common. 3. Select the lowest exponent for each variable 1) 4x3 and 6x7 4x3 = 2 · 2 · x · x · x 6x7 = 2 · 3 x · x · x · x · x · x · x So the GCF is 2 · x · x · x = 2x3 Martin-Gay, Developmental Mathematics 6 Greatest Common Factor GCF of monomials or terms – 1. write the gcf of the coefficients. 2. write the variables they have in common. 3. Select the lowest exponent for each variable all the integers or polynomials involved. 1) 2) 8y3z + 12y2z 8y3z = 2 · 2 · 2 · y · y · y · z 12y2 z = 2 · 2 · 3 ·y · y · z So the GCF is 2 · 2 · y · y · z = 4y2 z Martin-Gay, Developmental Mathematics 7 Factoring Polynomials The first step in factoring a polynomial is to find the GCF of all its terms. Then we write the polynomial as a product by factoring out the GCF from all the terms. Martin-Gay, Developmental Mathematics 8 Factoring out the GCF Factor out the GCF in each of the following polynomials. 1) 6x3 – 9x2 + 12x = 1 3x ( 2x2 –3x + 4 ) = 6x3 – 9x2 + 12x GCF what is the largest number that divides evenly into 6, 9 & 12? Is there a variable that each term has in common? what is the smallest exponent for the x of all 3 terms? That is the GCF Now open a set of parentheses and divide the polynomial by the GCF to find the other factor Check by multiplying the GCF and the other factor Martin-Gay, Developmental Mathematics 9 Factoring out the GCF Factor out the GCF in each of the following polynomial. 2) 14x4 + 7x3 – 21x2 = 2 7x ( 2x2 + x – 3 ) = 14x4 + 7x3 – 21x2 GCF what is the largest number that divides evenly into 14, 7 & 21? Is there a variable that each term has in common? what is the smallest exponent for the x of all 3 terms? That is the GCF Now open a set of parentheses and divide the polynomial by the GCF to find the other factor Check by multiplying the GCF and the other factor Martin-Gay, Developmental Mathematics 10 Factoring Remember that factoring out the GCF from the terms of a polynomial should always be the first step in factoring a polynomial. This will usually be followed by additional steps in the process. Martin-Gay, Developmental Mathematics 11 Factoring Trinomials of the 2 Form x + bx + c Factoring Trinomials Recall by using the FOIL method that F O I L (x + 2)(x + 4) = x2 + 4x + 2x + 8 = x2 + 6x + 8 To factor x2 + bx + c into (x + one #)(x + another #), note that b is the sum of the two numbers and c is the product of the two numbers. So we’ll be looking for 2 numbers whose product is c and whose sum is b. Note: there are fewer choices for the product, so that’s why we start there first. Martin-Gay, Developmental Mathematics 13 Factoring Polynomials Example ax bx c 2 Factor the polynomial 1. x2 + 13x + 30. 30 3 10 Factor out GCF 1 2. Multiply a c and put at top of he X 13 1 3. Put b at the bottom of X 4. Find the pair of factors of 30 that add to 13 5. Put each of the factors over a Positive Factors of 30 (DenominatorX Sum of factors 6. Write the factors. + numerator) 1 30 31 (1x 3)(1x 10) 2 15 17 3 10 13 2 1. ( x 3)( x 10) x 13x 30 Martin-Gay, Developmental Mathematics 14 Factoring Polynomials ax bx c 2 Example Factor the polynomial 2. x2 – 11x + 24. 24 -3 -8 Factor out GCF 1 2. Multiply and put at top of he X -11 1 3. Put b at the bottom of X 4. Find the pair of factors of 24 that add to -11 5. Put each of the factors over 6. Write the factors. (DenominatorX + numerator) Factors of 24 Sum of factors (1x 3)(1x 8) -1 -24 -25 -2 -12 -14 1. a c a -3 -8 2 ( x 3 )( x 8 ) x 11x 24 -11 Martin-Gay, Developmental Mathematics 15 Factoring Polynomials Example ax bx c 2 Factor the polynomial 3. x2 – 2x – 35. -35 5 -7 Factor out GCF 1 2. Multiply a c and put at top of he X -2 1 3. Put b at the bottom of X 4. Find the pair of factors of -35 that add to -2 5. Put each of the factors over a Positive Factors of -35(DenominatorX Sum of factors 6. Write the factors + numerator) -1 35 34 (1x 5)(1x 7) 1 -35 -34 -5 7 2 2 5 -7 -2 ( x 5)( x 7) x 2 x 35 1. Martin-Gay, Developmental Mathematics 16 Prime Polynomials Example Factor the polynomial 4. x2 – 6x + 10. Since our two numbers must have a product of 10 and a sum of – 6, the two numbers will have to both be negative. Negative factors of 10 Sum of Factors – 1, – 10 – 11 – 2, – 5 –7 Since there is not a factor pair whose sum is – 6, x2 – 6x +10 is not factorable and we call it a prime polynomial. Martin-Gay, Developmental Mathematics 17 Check Your Result! You should always check your factoring results by multiplying the factored polynomial to verify that it is equal to the original polynomial. Many times you can detect computational errors or errors in the signs of your numbers by checking your results. Martin-Gay, Developmental Mathematics 18 Factoring Trinomials of 2 the Form ax + bx + c Factoring Trinomials Returning to the FOIL method, F O I L (3x + 2)(x + 4) = 3x2 + 12x + 2x + 8 = 3x2 + 14x + 8 To factor ax2 + bx + c into (#1·x + #2)(#3·x + #4), note that a is the product of the two first coefficients, c is the product of the two last coefficients and b is the sum of the products of the outside coefficients and inside coefficients. Note that b is the sum of 2 products, not just 2 numbers, as in the last section. Martin-Gay, Developmental Mathematics 20 Factoring Polynomials Example ax bx c 2 Factor the polynomial 7. 3x2 + 5x + 2. 6 2 3 1 Factor out GCF 3 3 1 2. Multiply a c and put at top of he X 5 3. Put b at the bottom of X 4. Find the pair of factors of 6 that add to 5 5. Put each of the factors over a 6. Write (DenominatorX + numerator) Factors of the 6 factors Sum of factors 1 6 7 (3x 2)(1x 1) 2 3 5 -1 -6 -7 2 ( 3 x 2 )( x 1 ) 3 x 5x 2 -2 -3 -5 1. Martin-Gay, Developmental Mathematics 21 Factoring Polynomials Example ax bx c 2 Factor the polynomial 8. 5d2 + 6d – 8. -40 10 2 -4 Factor out GCF 5 1 5 2. Multiply a c and put at top of he X 6 3. Put b at the bottom of X 4. Find the pair of factors of -40 that add to 6 5. Put each of the factors over a Factors of the - 40factors (DenominatorX Sum of factors+ numerator) 6. Write 1 -40 -39 2 -20 -18 (5d 4)( d 2) 4 -10 -6 2 ( 5 d 4 )( d 2 ) 5 d 6d 8 -4 10 6 1. Martin-Gay, Developmental Mathematics 22