Chapter 9.1 Factoring a number

Objective NCSCOS 1.01 – Write equivalent forms of algebraic expressions to solve problems Students will know how to factor number and find the greatest common factor

Vocabulary Factoring a number Prime number – Composite number – A number that can only be divided by 1 and itself Ex. 2, 3, 5, 7, 11, 13 Composite number – A number that can be divided by more than 1 and itself Ex. 4: (1*4, 2*2) Ex. 6: (1*6, 2*3)

Vocabulary Factoring a number Prime Factorization – A composite number expressed as a product of factors that are all prime numbers The process of factoring a number down to only prime numbers

Factoring a number Label the following as prime or composite 24 23 18 9 53

Factoring a number Label the following as prime or composite 24 23 18 9 53 Composite Prime Composite Composite Prime

Factoring a number Example 1: Find the prime factorization of 90 90 The smallest prime number is 2, so we start with 2 When you take a 2 out, you’re left with 45 2 45

Factoring a number Example 1: Find the prime factorization of 90 90 The two numbers below 90 should multiply to the number above it 2 can’t be factored any more, so we’ll have to factor the 45 more 2 45 x = 90

Factoring a number Example 1: Find the prime factorization of 90 90 The next smallest prime number is 3, so take 3 out of 45 Notice that 3 and 15 multiply to the number above it 2 45 3 15 x = 45

Factoring a number Example 1: Find the prime factorization of 90 90 15 can be factored by 3 again At this point, all the number are prime 2 45 3 15 3 5

90 2 45 3 15 * 3 * 3 * 5 = 90 2 3 5 Factoring a number Example 1: Find the prime factorization of 90 90 We can multiply all these numbers together to get 90 2 45 3 15 * 3 * 3 * 5 = 90 2 3 5

90 2 45 3 15 * 3 * 32 * 3 * 5 = 90 2 3 5 Factoring a number Example 1: Find the prime factorization of 90 90 We can then write 3 * 3 as 32 We now have 90 fully factored 2 45 3 15 * 3 * 32 * 3 * 5 = 90 2 3 5

-28 -1 28 2 14 2 7 -1 * 22 * 2 * 2 * 7 Factoring a number Example 2: Factor -28 If it’s negative, factor out -1 first Now factor a positive 28 -1 28 2 14 2 7 -1 * 22 * 2 * 2 * 7

Factoring a number Factor the following numbers: 1. 24 2. 36 3. -110 4. 168 5. 210

Factoring a number Factor the following numbers: 1. 24 2. 36 23 * 3 22 * 32 3. -110 4. 168 -1 * 2 * 5 * 11 23 * 3 * 7 5. 210 2 * 3 * 5 * 7

36x3 2 18x3 2 9x3 3 3x3 2 * 2 * 3 * 3x3 Factoring a number Example 3: Factor 2 Factor the number first 18x3 2 9x3 3 3x3 2 * 2 * 3 * 3x3

36x3 2 18x3 2 9x3 3 3x3 2 * 2 * 3 * 3 * 3x3 *x*x*x Factoring a number Example 3: Factor 2 Then factor the variable Remember, x3 is x*x*x 18x3 2 9x3 3 3x3 2 * 2 * 3 * 3 * 3x3 *x*x*x

36x3 22 * 32 * x3 2 * 2 * 3 * 3 * x * x * x Factoring a number Example 3: Factor We can now reduce this to get our answer 22 * 32 * x3 2 * 2 * 3 * 3 * x * x * x

Factoring a number 1. 20x2 2. 32x5 3. -120x3 4. 68x3y5 5. 252x4y2 Factor the following: 1. 20x2 2. 32x5 3. -120x3 4. 68x3y5 5. 252x4y2

Factoring a number 1. 20x2 2. 32x5 22 * 5 * x2 25 * x5 3. -120x3 Factor the following: 1. 20x2 2. 32x5 22 * 5 * x2 25 * x5 3. -120x3 4. 68x3y5 -1 * 23 * 3 * 5 * x3 22 * 17 * x3 * y5 5. 252x4y2 22 * 33 * 7 * x4 *y2

Factoring a number Example 4:Find the Greatest Common Factor of : 24 36 and We need to factor both numbers first

Factoring a number 24 Factor 24 2 12 2 6 2 3

Factoring a number 36 Factor 36 2 18 9 2 3 3

Factoring a number Write out each numbers factors Find what they have in common Multiply what they have in common to find the Greatest Common Factor GCF 24 = 2 * 2 * 2 * 3 36 = 2 * 2 * 3 * 3 2 * 2 * 3 = 12

28x3 42x5 28x3 = 2*2*7*x*x*x 42x5 = 2*3*7*x*x*x*x*x GCF = 2*7*x*x*x Factoring a number 28x3 42x5 Find the GCF for: Factor each number first Find what they have in common and 28x3 = 2*2*7*x*x*x 42x5 = 2*3*7*x*x*x*x*x GCF = 2*7*x*x*x = 2*7*x3

28x3 42x5 28x3 = 2*2*7*x*x*x 42x5 = 2*3*7*x*x*x*x*x GCF = 2*7*x3 Factoring a number 28x3 42x5 Find the GCF for: Factor each number first Find what they have in common and 28x3 = 2*2*7*x*x*x 42x5 = 2*3*7*x*x*x*x*x GCF = 2*7*x3 = 14x3

28x3 42x5 GCF = 14x3 Factoring a number and Find the GCF for: You have to factor the numbers to find the GCF but you can use a shortcut for the variables: Which value of x has the lowest number as an exponent will be your GCF and GCF = 14x3

Factoring a number 1. 12 and 16 2. 120 and 180 4. 52x4 and 104x4 Find the Greatest Common Factor for the following: 1. 12 and 16 2. 120 and 180 4. 52x4 and 104x4 3. 54x3 and 90x5 5. 312x5 and 468x7

Factoring a number 1. 12 and 16 2. 120 and 180 4 60 4. 52x4 and 104x4 Find the Greatest Common Factor for the following: 1. 12 and 16 2. 120 and 180 4 60 4. 52x4 and 104x4 3. 54x3 and 90x5 18x3 26x4 5. 312x5 and 468x7 156x5

Factoring a number Example: Find the GCF for the following numbers: 12x3 18x2 42x5 First, factor out each number 12x3 = 2 * 2 * 3 * x3 18x2 = 2 * 3 * 3 * x2 42x5 = 2 * 3 * 7 * x5

Factoring a number Find what they have in common They all have one 2 They also have one 3 * 3 2 12x3 = 2 * 2 * 3 * x3 18x2 = 2 * 3 * 3 * x2 42x5 = 2 * 3 * 7 * x5

Factoring a number Multiply the numbers Find the smallest value of x and plug it in * 3 2 = 6 x2 12x3 = 2 * 2 * 3 * x3 18x2 = 2 * 3 * 3 * x2 42x5 = 2 * 3 * 7 * x5

Quiz! 12x3 18x2 42x5 6x2 Factoring a number Example: Find the GCF for the following numbers: Quiz! 12x3 18x2 42x5 6x2 The answer is:

Quiz! Factoring a number Factor: 252 Factor: -594x2y4 Find the greatest common factor: 48 and 120 84x5 and 112x3 117x7, 468x4 and 351x5

Quiz! Factoring a number Factor: 252 Factor: -594x2y4 Find the greatest common factor: 48 and 120 84x5 and 112x3 117x7, 468x4 and 351x5 22*32*7 -1*2*33*11*x2*y4 24 28x3 117x4