Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematics of radio engineering wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Horner's method wikipedia , lookup
Vincent's theorem wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
System of polynomial equations wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Elementary mathematics wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
MATH 1000 /11 Chapter 6 6. 1. Greatest Common Factor and Factoring by Grouping 6. 2. Factoring Trinomials of the form a x2 + bx + c 5/24/2017 1 • Factors (either numbers or polynomials) • When an integer is written as a product of integers, each of the integers in the product is a factor of the original number. • When a polynomial is written as a product of polynomials, each of the polynomials in the product is a factor of the original polynomial. • Factoring – writing a polynomial as a product of polynomials. 5/24/2017 2 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. 5/24/2017 3 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. 5/24/2017 4 Example Find the GCF of each list of numbers. 5/24/2017 1) 6, 8 and 46 6=2•3 8=2•2•2 46 = 2 • 23 So the GCF is 2. 2) 144, 256 and 300 144 = 2 • 2 • 2 • 3 • 3 256 = 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 300 = 2 • 2 • 3 • 5 • 5 So the GCF is 2 • 2 = 4. 5 Example Find the GCF of each list of terms. x3 and x7 x3 = x • x • x x7 = x • x • x • x • x • x • x So the GCF is x • x • x = x3 2) 6x5 and 4x3 6x5 = 2 • 3 • x • x • x • x • x 4x3 = 2 • 2 • x • x • x So the GCF is 2 • x • x • x = 2x3 1) 5/24/2017 6 Example Find the GCF of the following list of terms. a3b2, a2b5 and a4b7 a3b2 = a • a • a • b • b a2b5 = a • a • b • b • b • b • b a4b7 = a • a • a • a • b • b • b • b • b • b • b So the GCF is a • a • b • b = a2b2 Notice that the GCF of terms containing variables will use the smallest exponent found amongst the individual terms for each variable. 5/24/2017 7 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. The remaining factors in each term will form a polynomial. 5/24/2017 8 Example Factor out the GCF in each of the following polynomials. 1) 6x3 – 9x2 + 12x = 3 • x • 2 • x2 – 3 • x • 3 • x + 3 • x • 4 = 3x(2x2 – 3x + 4) 2) 14x3y + 7x2y – 7xy = 7 • x • y • 2 • x2 + 7 • x • y • x – 7 • x • y • 1 = 7xy(2x2 + x – 1) 5/24/2017 9 Example Factor out the GCF in each of the following polynomials. 1) 6(x + 2) – y(x + 2) = 6 • (x + 2) – y • (x + 2) = (x + 2)(6 – y) 2) xy(y + 1) – (y + 1) = xy • (y + 1) – 1 • (y + 1) = (y + 1)(xy – 1) 5/24/2017 10 Factoring polynomials often involves additional techniques after initially factoring out the GCF. One technique is factoring by grouping. Example Factor xy + y + 2x + 2 by grouping. Notice that, although 1 is the GCF for all four terms of the polynomial, the first 2 terms have a GCF of y and the last 2 terms have a GCF of 2. xy + y + 2x + 2 = x • y + 1 • y + 2 • x + 2 • 1 = y(x + 1) + 2(x + 1) = 5/24/2017 (x + 1)(y + 2) 11 Factoring a four-term Polynomial by Grouping 1) Arrange the terms so that the first two terms have a common factor and the last two terms have a common factor. 2) For each pair of terms, use the distributive property to factor out the pair’s greatest common factor. 3) If there is now a common binomial factor, factor it out. 4) If there is no common binomial factor in step 3, begin again, rearranging the terms differently. • 5/24/2017 If no rearrangement leads to a common binomial factor, the polynomial cannot be factored. 12 Example Factor each of the following polynomials by grouping. 1) x3 + 4x + x2 + 4 = x • x2 + x • 4 + 1 • x2 + 1 • 4 = x(x2 + 4) + 1(x2 + 4) = (x2 + 4)(x + 1) 2) 2x3 – x2 – 10x + 5 = x2 • 2x – x2 • 1 – 5 • 2x – 5 • (-1) = x2(2x – 1) – 5(2x – 1) = (2x – 1)(x2 – 5) 5/24/2017 13 Example Factor 2x – 9y + 18 – xy by grouping. Neither pair has a common factor (other than 1). So, rearrange the order of the factors. 2x + 18 – 9y – xy = 2 • x + 2 • 9 – 9 • y – x • y = 2(x + 9) – y(9 + x) = 2(x + 9) – y(x + 9) = (make sure the factors are identical) (x + 9)(2 – y) 5/24/2017 14 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. Example Factor 90 + 15y2 – 18x – 3xy2. 90 + 15y2 – 18x – 3xy2 = 3(30 + 5y2 – 6x – xy2) = 3(5 • 6 + 5 • y2 – 6 • x – x • y2) = 3(5(6 + y2) – x (6 + y2)) = 5/24/2017 3(6 + y2)(5 – x) 15 Section 6.2 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. 5/24/2017 16 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). 5/24/2017 17 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). 5/24/2017 18 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). 5/24/2017 19 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 Now we have a problem, because we have exhausted all possible choices for the factors, but have not found a pair whose sum is -6. 2 – 6x +10 is not factorable and we call it a prime So x 5/24/2017 20 polynomial. Example Factor the polynomial x2 – 11xy + 30y2. We look for two terms whose product is 30y2 and whose sum is –11y. The two terms will have to both be negative. Note: each term will contain the variable y, for the sum to be –11y. Negative factors of 30y2 Sum of Factors -y, -30y -31y -2y, -15y -17y -3y, -10y -13y -5y, -6y -11y So x2 – 11xy + 30y2 = (x – 5y)(x – 6y). 5/24/2017 21 Example Factor the polynomial 3x6 + 30x5 + 72x4. First we factor out the GCF. 3x6 + 30x5 + 72x4 = 3x4(x2 + 10x + 24) Then we factor the trinomial. Positive factors of 24 Sum of Factors 1, 24 25 2, 12 14 3, 8 11 4, 6 10 So 3x6 + 30x5 + 72x4 = 3x4(x2 + 10x + 24) = 3x4(x + 4)(x + 6). 5/24/2017 22 • 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. 5/24/2017 23