mth 10905 algebra factoring a monomial from a polynomial chapter 5 section 1

MTH 10905Algebra

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 – 52x + (-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

Example: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 + 2xx and (x + 2) are factors of x2 + 2x

Example:(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

9 • x3

x • 9x2 3x • 3x2

9x • x2

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: Multiply7(x + 2) (7)(x) + (7)(2) 7x + 14

Example: Factoring7x + 14 7(x + 2)

Examples of Multiplying and Factoring

Example: Multiply2(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: Factoring6x2 – 10x – 4

2(x – 2)(3x + 1)

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: Factoringx2 – 9x + 20 (x – 5)(x – 4)

Determine the GCFof 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 GCFof 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 GCFof 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 GCFof 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 GCFof 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

1. Write each number as a product of prime factors.

2. Determine the prime factors common to all numbers.

3. 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 GCFof 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 GCFof 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 GCFof 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 GCFof 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 GCFof 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 GCFof 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 GCFof 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 Monomialfrom a Polynomial

Steps to Factor a Monomial from a Polynomial:

1. Determine the greatest common factor of all terms in the polynomial

2. Write each term as a product of the GCF and its other factors

3. 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 Monomialfrom 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 Monomialfrom 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 Monomialfrom a Polynomial

Example: Factor 8x5 + 12x2 – 44x GCF = 2x • 2x = 4x

8x5 + 12x2 – 44x = (4x • 2x4)+ (4x • 3x) – (4x • 11) = 4x(2x2 + 3x – 11)

Factor a Monomialfrom a Polynomial

Example: Factor 60p2 – 12p – 18 GCF = 2 • 3 = 6

60p2 – 12p – 18 = (6 • 10p2)– (6 • 2p) – (6 • 3) = 6(10p2 – 2p – 3)

Factor a Monomialfrom a Polynomial

Example: Factor 3x3 + x2 + 9x2y GCF = x2

3x3 + x2 + 9x2y = (x2 • 3x) + (x2 • 1) + (x2 • 9y) = x2(3x + 1 + 9y)

Factor a Monomialfrom a Polynomial

Example: Factor x(6x + 5) + 9(6x + 5) GCF = 6x + 5

x(6x + 5) + 9(6x + 5)= x • (6x + 5) + 9 • (6x + 5) = (6x+5)(x + 9)

Factor a Monomialfrom a Polynomial

Example: Factor3x(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 Monomialfrom a Polynomial

Example: Factor 6y(5y – 2) – 5(5y – 2)GCF = 5y – 2

6y(5y – 2) – 5(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

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