Factoring a Monomial from a Polynomial Chapter 5 Section 1 MTH 10905 Algebra Factoring a Monomial from a Polynomial Chapter 5 Section 1

Identify Factors Factor an expression means to write the expression as a product of its factors Factoring can be used to solve equations and perform operations on fractions. Factoring is the reverse process of multiplying.

Identify Factors Remember: A term is parts that are added For example: 2x – 3y – 5 2x + (-3y) + (-5) A factor is variables that are multiplied Therefore, if a • b = c then a and b are factors of c.

Identify Factors Example: 3 • 5 = 15 3 and 5 are factors of 15 x3 • x4 = x7 x3 and x4 are factors of x7 We general list only the positive factors, however, the negatives or opposites of each of these are also factors.

Identify Factors Example: x(x+2) = x2 + 2x x and (x + 2) are factors of x2 + 2x (x – 1)(x + 3) = x2 + 2x -3 (x – 1) and (x + 3) are factors of x2 + 2x -3

Identify Factors Example: List the factors of 9x3 1 • 9x3 3 • 3x3 Therefore: 1, 3, 9, x, 3x, 9x, x2, 3x2, 9x2, x3, 3x3, 9x3 and the opposites of these are factors of 9x3

Examples of Multiplying and Factoring Example: Multiply 7(x + 2) (7)(x) + (7)(2) 7x + 14 Example: Factoring

Examples of Multiplying and Factoring Example: Multiply 2(x – 2)(3x + 1) 2[(x)(3x)+(x)(1)+(-2)(3x)+(-2)(1)] (2)(x)(3x)+(2)(x)(1)+(2)(-2)(3x)+(2)(-2)(1) 6x1+1 + 2x – 12x – 4 6x2 – 10x – 4 Example: Factoring

Examples of Multiplying and Factoring Example: Multiply (x – 5)(x – 4) (x)(x) + (x)(-4) + (-5)(x) + (-5)(-4) x1+1 – 4x – 5x + 20 x2 – 9x + 20 Example: Factoring

Determine the GCF of Two or More Numbers To factor we need to make use the Greatest Common Factor (GCF). If you are having trouble seeing the GCF you can start with a common factor and continuing pulling out the common factors until no common factors remain. Remember that the GCF of two or more numbers is the greatest number that divides into all the numbers Example: GCF of 6 and 8 is 2

Determine the GCF of Two or More Numbers When the GCF is not easy to find we can find it by writing each number as a product of prime numbers. Prime Number is an integer greater than 1 that has exactly two factors, itself and one. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Determine the GCF of Two or More Numbers Positive integers greater than 1 that are not prime are called composite numbers. The first 15 composite numbers are: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25 All even number greater than 2 are composite numbers. The number 1 is called a unit. One is not a prime number and it is not a composite number.

Determine the GCF of Two or More Numbers Example: Write 54 as a product of prime numbers. 54 = 2 • 3 • 3 • 3 = 2 • 33 6 9 2 3 3 3 Prime Factorization of 54

Determine the GCF of Two or More Numbers Example: Write 80 as a product of its prime factors. 80 = 2 • 2 • 2 • 2 • 5 = 24 • 5 8 10 2 4 2 5 2 2 2 2 5 Prime Factorization of 80

Determine the GCF of Two or More Numbers Write each number as a product of prime factors. Determine the prime factors common to all numbers. Multiply the common factors to get the GCF

Determine the GCF of Two or More Numbers Example: Determine the GCF of 48 and 80. 48 80 (6) (8) (8) (10) (2)(3) (2)(4) (2)(4) (2)(5) (2)(3) (2)(2)(2) (2)(2)(2) (2)(5) 2 • 3 • 2 • 2 • 2 24 • 3 2 • 2 • 2 • 2 • 5 24 • 5 GCF = 24 = 16

Determine the GCF of Two or More Numbers Example: Determine the GCF of 56 and 124. 56 124 (2) (28) (2) (62) (2) (2)(14) (2) (2)(31) (2) (2)(2)(7) 2 • 2 • 2 • 7 2 • 2 • 31 23 • 7 22 • 31 GCF = 22 = 4

Determine the GCF of Two or More Terms Example: Determine the GCF of the terms: y8, y2, y6, and y10 To determine the GCF of two or more terms, take each factor the largest number of times that it appears in all the terms. y8 = y2 • y2 y2 = y2 • 1 GCF = y2 y6 = y2 • y4 y10 = y2 • y8

Determine the GCF of Two or More Terms Example: Determine the GCF of the terms: a2b7, a4b, and a8b2 a2b7 = a2 • b • b6 a4b = a2 • a2 • b a8b2 = a2 • a6 • b • b GCF = a2b

Determine the GCF of Two or More Terms Example: Determine the GCF of the terms: pq, p3q, and q2 pq = p • q p3q = p • p2 • q q2 = q • q GCF = q

Determine the GCF of Two or More Terms Example: Determine the GCF of the terms. -12b3, 18b2, and 28b -12b3 = -1 • 2 • 2 • 3 • b • b2 18b2 = 2 • 3 • 3 • b • b 28b = 2 • 2 • 7 • b GCF = 2b

Determine the GCF of Two or More Terms Example: Determine the GCF of the terms. y3, 9y5, and y2 y3 = y • y2 9y5 = 9 • y2 • y3 y2 = y2 GCF = y2

Determine the GCF of Two or More Terms Example: Determine the GCF of the pair of terms. y(y - 2) and 3(y – 2) y(y – 2) = y • (y – 2) 3(y – 2) = 3 • (y – 2) GCF = (y – 2)

Determine the GCF of Two or More Terms Example: Determine the GCF of the pair of terms. 3(x + 6) and x + 6 3(x + 6) = 3 • (x + 6) 1(x + 6) = 1 • (x + 6) GCF = (x + 6)

Factor a Monomial from a Polynomial Steps to Factor a Monomial from a Polynomial: Determine the greatest common factor of all terms in the polynomial Write each term as a product of the GCF and its other factors Use the distributive property to factor out the GCF Example: Factor 8y + 12 GCF = 2 • 2 = 4 8y + 12 = (4 • 2y) + (4 • 3) = 4(2y + 3)

Factor a Monomial from a Polynomial Example: Factor 24x – 18 GCF = 6 24x – 18 = (6 • 4x) – (6 • 3) = 6(4x – 3) To check the factoring process, multiply the factors using the distributive property. If the factoring is correct, the product will be the polynomial you start with.

Factor a Monomial from a Polynomial Example: Factor 8w2 + 12w6 GCF = 2w • 2w = 4w2 8w2 + 12w6 = (4w2 • 2) + (4w2 • 3w4) = 4w2(2 + 3w4) Check: 4w2 (2 + 3w4) (4w2)(2) + (4w2)(3w4) 8w2 + 12w6

Factor a Monomial from a Polynomial Example: Factor 8x5 + 12x2 – 44x GCF = 2x • 2x = 4x = (4x • 2x4)+ (4x • 3x) – (4x • 11) = 4x(2x2 + 3x – 11)

Factor a Monomial from a Polynomial Example: Factor 60p2 – 12p – 18 GCF = 2 • 3 = 6 = (6 • 10p2)– (6 • 2p) – (6 • 3) = 6(10p2 – 2p – 3)

Factor a Monomial from a Polynomial Example: Factor 3x3 + x2 + 9x2y GCF = x2 = (x2 • 3x) + (x2 • 1) + (x2 • 9y) = x2(3x + 1 + 9y)

Factor a Monomial from a Polynomial Example: Factor x(6x + 5) + 9(6x + 5) GCF = 6x + 5 = x • (6x + 5) + 9 • (6x + 5) = (6x+5)(x + 9)

Factor a Monomial from a Polynomial Example: Factor 3x(x – 3) – 2(x – 3) GCF = x – 3 3x(x – 3) – 2(x – 3) = 3x • (x – 3) – 2 • (x – 3) = (x – 3)(3x –2)

Factor a Monomial from a Polynomial Example: Factor 6y(5y – 2) – 5(5y – 2) GCF = 5y – 2 = 6y • (5y – 2) – 5 • (5y – 2) = (5y – 2)(6y – 5)

IMPORTANT Whenever you are factoring a polynomial by any method; the first step is to see if there are any common factors (other than 1) to all the terms in the polynomial. If yes, factor the GCF from each term using the distributive property.

HOMEWORK 5.1 Page 298: #49, 51, 55, 61, 69, 79, 81, 89, 91