University:
California State University, Northridge
Course:
MATH 150A | Calculus I
Academic year:

2024
Views:

420

Pages:

25
Author:

Shelby Rollins

Chapter 13 Factoring Polynomials § 13.2 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. 15 Factoring Polynomials Example Factor the polynomial x2 + 13x + 30. Since our two numbers must have a product of 30 and a sum of 13, the two numbers must both be positive. Positive factors of 30 Sum of Factors 1, 30 31 2, 15 17 3, 10 13 Note, there are other factors, but once we find a pair that works, we do not have to continue searching. So x2 + 13x + 30 = (x + 3)(x + 10). 16 Factoring Polynomials Example Factor the polynomial x2 – 11x + 24. Since our two numbers must have a product of 24 and a sum of -11, the two numbers must both be negative. Negative factors of 24 Sum of Factors – 1, – 24 – 25 – 2, – 12 – 14 – 3, – 8 – 11 So x2 – 11x + 24 = (x – 3)(x – 8). 17 Factoring Polynomials Example Factor the polynomial x2 – 2x – 35. Since our two numbers must have a product of – 35 and a sum of – 2, the two numbers will have to have different signs. Factors of – 35 Sum of Factors – 1, 35 34 1, – 35 – 34 – 5, 7 2 5, – 7 –2 So x2 – 2x – 35 = (x + 5)(x – 7). 18 Prime Polynomials Example Factor the polynomial 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. 19 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. 20 § 13.3 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. 22 Factoring Polynomials Example Factor the polynomial 25x2 + 20x + 4. Possible factors of 25x2 are {x, 25x} or {5x, 5x}. Possible factors of 4 are {1, 4} or {2, 2}. We need to methodically try each pair of factors until we find a combination that works, or exhaust all of our possible pairs of factors. Keep in mind that, because some of our pairs are not identical factors, we may have to exchange some pairs of factors and make 2 attempts before we can definitely decide a particular pair of factors will not work. Continued. 23 Factoring Polynomials Example Continued We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to 20x. Product of Product of Sum of Factors Factors Resulting of 25x2 of 4 Binomials Outside Terms Inside Terms Products {x, 25x} {1, 4} (x + 1)(25x + 4) 4x 25x 29x (x + 4)(25x + 1) x 100x 101x {x, 25x} {2, 2} (x + 2)(25x + 2) 2x 50x 52x {5x, 5x} {2, 2} (5x + 2)(5x + 2) 10x 10x 20x Continued. 24 Factoring Polynomials Example Continued Check the resulting factorization using the FOIL method. F O I L (5x + 2)(5x + 2) = 5x(5x) + 5x(2) + 2(5x) + 2(2) = 25x2 + 10x + 10x + 4 = 25x2 + 20x + 4 So our final answer when asked to factor 25x2 + 20x + 4 will be (5x + 2)(5x + 2) or (5x + 2)2. 25 Factoring Polynomials Example Factor the polynomial 21x2 – 41x + 10. Possible factors of 21x2 are {x, 21x} or {3x, 7x}. Since the middle term is negative, possible factors of 10 must both be negative: {-1, -10} or {-2, -5}. We need to methodically try each pair of factors until we find a combination that works, or exhaust all of our possible pairs of factors. Continued. 26 Factoring Polynomials Example Continued We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to −41x. Factors Factors Resulting Product of Product of Sum of of 21x2 of 10 Binomials Outside Terms Inside Terms Products {x, 21x}{1, 10}(x – 1)(21x – 10) –10x −21x – 31x (x – 10)(21x – 1) –x −210x – 211x {x, 21x} {2, 5} (x – 2)(21x – 5) –5x −42x – 47x (x – 5)(21x – 2) –2x −105x – 107x Continued. 27 Factoring Polynomials Example Continued Factors Factors Resulting Product of Product of Sum of of 21x2 of 10 Binomials Outside Terms Inside Terms Products {3x, 7x}{1, 10}(3x – 1)(7x – 10) −30x −7x −37x (3x – 10)(7x – 1) −3x −70x −73x {3x, 7x} {2, 5} (3x – 2)(7x – 5) −15x −14x −29x (3x – 5)(7x – 2) −6x −35x −41x Continued. 28 Factoring Polynomials Example Continued Check the resulting factorization using the FOIL method. F O I L (3x – 5)(7x – 2) = 3x(7x) + 3x(-2) - 5(7x) - 5(-2) = 21x2 – 6x – 35x + 10 = 21x2 – 41x + 10 So our final answer when asked to factor 21x2 – 41x + 10 will be (3x – 5)(7x – 2). 29 Factoring Polynomials Example Factor the polynomial 3x2 – 7x + 6. The only possible factors for 3 are 1 and 3, so we know that, if factorable, the polynomial will have to look like (3x )(x ) in factored form, so that the product of the first two terms in the binomials will be 3x2. Since the middle term is negative, possible factors of 6 must both be negative: {−1, − 6} or {− 2, − 3}. We need to methodically try each pair of factors until we find a combination that works, or exhaust all of our possible pairs of factors. Continued. 30 Factoring Polynomials Example Continued We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to −7x. Factors of 6 Resulting Binomials {−1, −6} (3x – 1)(x – 6) (3x – 6)(x – 1) {−2, −3} (3x – 2)(x – 3) (3x – 3)(x – 2) Product of Product of Sum of Outside Terms Inside Terms Products −18x −x −19x Common factor so no need to test. −9x −2x −11x Common factor so no need to test. Continued. 31 Factoring Polynomials Example Continued Now we have a problem, because we have exhausted all possible choices for the factors, but have not found a pair where the sum of the products of the outside terms and the inside terms is –7. So 3x2 – 7x + 6 is a prime polynomial and will not factor. 32 Factoring Polynomials Example Factor the polynomial 6x2y2 – 2xy2 – 60y2. Remember that the larger the coefficient, the greater the probability of having multiple pairs of factors to check. So it is important that you attempt to factor out any common factors first. 6x2y2 – 2xy2 – 60y2 = 2y2(3x2 – x – 30) The only possible factors for 3 are 1 and 3, so we know that, if we can factor the polynomial further, it will have to look like 2y2(3x )(x ) in factored form. Continued. 33 Factoring Polynomials Example Continued Since the product of the last two terms of the binomials will have to be –30, we know that they must be different signs. Possible factors of –30 are {–1, 30}, {1, –30}, {–2, 15}, {2, –15}, {–3, 10}, {3, –10}, {–5, 6} or {5, –6}. We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to –x. Continued. 34 Factoring Polynomials Example Continued Factors Resulting Product of Product of Sum of of -30 Binomials Outside Terms Inside Terms Products {-1, 30} (3x – 1)(x + 30) 90x -x 89x (3x + 30)(x – 1) {1, -30} (3x + 1)(x – 30) (3x – 30)(x + 1) {-2, 15} (3x – 2)(x + 15) (3x + 15)(x – 2) {2, -15} (3x + 2)(x – 15) (3x – 15)(x + 2) Common factor so no need to test. -90x x -89x Common factor so no need to test. 45x -2x 43x Common factor so no need to test. -45x 2x -43x Common factor so no need to test. Continued. 35 Factoring Polynomials Example Continued Factors of –30 {–3, 10} Resulting Binomials (3x – 3)(x + 10) (3x + 10)(x – 3) {3, –10} (3x + 3)(x – 10) (3x – 10)(x + 3) Product of Product of Sum of Outside Terms Inside Terms Products Common factor so no need to test. –9x 10x x Common factor so no need to test. 9x –10x –x Continued. 36 Factoring Polynomials Example Continued Check the resulting factorization using the FOIL method. F O I L (3x – 10)(x + 3) = 3x(x) + 3x(3) – 10(x) – 10(3) = 3x2 + 9x – 10x – 30 = 3x2 – x – 30 So our final answer when asked to factor the polynomial 6x2y2 – 2xy2 – 60y2 will be 2y2(3x – 10)(x + 3). 37

Factoring Polynomials. Factoring Trinomials

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