lesson 1: factoring using the greatest common factor
factor each expression by factoring out the gcf.
1.) ( xy - xz )
2.) ( 9x^2 - 3x )
3.) ( 21b - 15a )
4.) ( 27y^3 + 18y^2 )
5.) ( 12x^2 - 16x )
6.) ( 28x^5 - 7x^2 )
7.) ( 2x^2y - 2xy )
8.) ( 8m^3 + 16m^2n )
9.) ( 4b^3 + 2b^2 + 8b )

Question

Accepted Answer

### Problem 1: $ xy - xz $
# Explanation:
## Step 1: Identify GCF
The terms $ xy $ and $ xz $ have a common factor of $ x $.
## Step 2: Factor out GCF
Factor $ x $ from both terms: $ x(y - z) $ (Note: The original handwritten answer had a sign error; it should be $ y - z $ instead of $ y + z $)
# Answer: $ x(y - z) $

### Problem 2: $ 9x^2 - 3x $ (Assuming the original $ 9x^4 - 3x $ might be a typo, or using $ 9x^2 - 3x $ as per the handwritten work)
# Explanation:
## Step 1: Identify GCF
The terms $ 9x^2 $ and $ 3x $ have a GCF of $ 3x $.
## Step 2: Factor out GCF
Divide each term by $ 3x $: $ 3x(3x - 1) $
# Answer: $ 3x(3x - 1) $

### Problem 3: $ 21b - 15a $
# Explanation:
## Step 1: Identify GCF
The GCF of 21 and 15 is 3.
## Step 2: Factor out GCF
Factor 3 from both terms: $ 3(7b - 5a) $
# Answer: $ 3(7b - 5a) $

### Problem 4: $ 27y^3 + 18y^2 $
# Explanation:
## Step 1: Identify GCF
The GCF of 27 and 18 is 9, and the GCF of $ y^3 $ and $ y^2 $ is $ y^2 $. So overall GCF is $ 9y^2 $? Wait, no: GCF of 27 and 18 is 9? Wait, 27 and 18: GCD(27,18)=9? No, GCD(27,18)=9? Wait, 27=3×9, 18=2×9, so GCD is 9? Wait, no, 27 and 18: GCD is 9? Wait, 27÷9=3, 18÷9=2. But also, the variables: $ y^3 $ and $ y^2 $ have GCF $ y^2 $. Wait, no, the GCF of the entire terms: for coefficients, GCD(27,18)=9? Wait, 27 and 18: GCD is 9? Wait, 27=3³, 18=2×3², so GCD is 3²=9? Wait, no, 3×9=27, 2×9=18, so yes. But also, the variables: $ y^3 $ and $ y^2 $ have GCF $ y^2 $. Wait, no, the GCF of the two terms is the product of the GCF of the coefficients and the GCF of the variables. So GCF of coefficients 27 and 18 is 9? Wait, no, 27 and 18: GCD is 9? Wait, 27 ÷ 9 = 3, 18 ÷ 9 = 2. But also, 27 and 18: GCD is 9? Wait, 3×9=27, 2×9=18. Alternatively, GCD(27,18)=9? Wait, no, 27 and 18: GCD is 9? Wait, 27=3×9, 18=2×9, so yes. But the variables: $ y^3 $ and $ y^2 $ have GCF $ y^2 $. So overall GCF is $ 9y^2 $? Wait, no, 27y³ = 9y²×3y, 18y²=9y²×2. So yes, GCF is $ 9y^2 $. Wait, but the handwritten work isn't shown here, but let's do it correctly.
## Step 1: Identify GCF
Coefficients: GCD(27,18)=9. Variables: GCF of $ y^3 $ and $ y^2 $ is $ y^2 $. So GCF is $ 9y^2 $.
## Step 2: Factor out GCF
$ 27y^3 + 18y^2 = 9y^2(3y + 2) $
Wait, but maybe the original problem was $ 27y^3 + 18y^2 $, and the GCF is 9y²? Wait, no, 27 and 18: GCD is 9? Wait, 27=3×9, 18=2×9. But 27 and 18: GCD is 9? Wait, 3×9=27, 2×9=18. Alternatively, 27 and 18: GCD is 9? Wait, 27 ÷ 9 = 3, 18 ÷ 9 = 2. So yes. But also, 27 and 18: GCD is 9? Wait, 3×9=27, 2×9=18. So GCF of coefficients is 9, and GCF of variables is $ y^2 $, so overall GCF is $ 9y^2 $. Then factoring out: $ 9y^2(3y + 2) $. But maybe the problem was $ 27y^3 + 18y^2 $ with GCF 9y²? Wait, no, 27 and 18: GCD is 9? Wait, 27=3³, 18=2×3², so GCD is 3²=9? Wait, 3×9=27, 2×9=18. So yes. So:
## Step 1: Find GCF of coefficients (27,18)=9; GCF of variables ($ y^3, y^2 $)=$ y^2 $. So GCF=9$ y^2 $.
## Step 2: Factor out 9$ y^2 $: $ 27y^3 + 18y^2 = 9y^2(3y + 2) $
# Answer: $ 9y^2(3y + 2) $ (If the original problem was $ 27y^3 + 18y^2 $; if it's $ 27y^3 + 18y^2 $ with GCF 9y²)

### Problem 5: $ 12x^2 - 16x $
# Explanation:
## Step 1: Identify GCF
The GCF of 12 and 16 is 4, and the GCF of $ x^2 $ and $ x $ is $ x $. So overall GCF is $ 4x $.
## Step 2: Factor out GCF
Divide each term by $ 4x $: $ 4x(3x - 4) $
# Answer: $ 4x(3x - 4) $

### Problem 6: $ 28x^5 - 7x^2 $
# Explanation:
## Step 1: Identify GCF
The GCF of 28 and 7 is 7, and the GCF of $ x^5 $ and $ x^2 $ is $ x^2 $. So overall GCF is $ 7x^2 $.
## Step 2: Factor out GCF
Divide each term by $ 7x^2 $: $ 7x^2(4x^3 - 1) $
# Answer: $ 7x^2(4x^3 - 1) $

### Problem 7: $ 2x^2y - 2xy $ (Assuming the original $ \frac{1}{2}x^2y - 2xy $ or a typo, using $ 2x^2y - 2xy $ as per the handwritten work)
# Explanation:
## Step 1: Identify GCF
The GCF of 2 and 2 is 2, and the GCF of $ x^2y $ and $ xy $ is $ xy $. So overall GCF is $ 2xy $.
## Step 2: Factor out GCF
Divide each term by $ 2xy $: $ 2xy(x - 1) $ (Matching the handwritten answer's corrected form)
# Answer: $ 2xy(x - 1) $

### Problem 8: $ 8m^3 + 16m^2n $
# Explanation:
## Step 1: Identify GCF
The GCF of 8 and 16 is 8, and the GCF of $ m^3 $ and $ m^2n $ is $ m^2 $. So overall GCF is $ 8m^2 $.
## Step 2: Factor out GCF
Divide each term by $ 8m^2 $: $ 8m^2(m + 2n) $
# Answer: $ 8m^2(m + 2n) $

### Problem 9: $ 4b^3 + 2b^2 + 8b $
# Explanation:
## Step 1: Identify GCF
The GCF of 4, 2, and 8 is 2, and the GCF of $ b^3, b^2, b $ is $ b $. So overall GCF is $ 2b $.
## Step 2: Factor out GCF
Divide each term by $ 2b $: $ 2b(2b^2 + b + 4) $
# Answer: $ 2b(2b^2 + b + 4) $

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( xy - xz \)

Step 1: Identify GCF

Step 2: Factor out GCF

Step 1: Identify GCF

Step 2: Factor out GCF

Step 1: Identify GCF

Step 2: Factor out GCF

Answer:

Problem 2: \( 9x^2 - 3x \) (Assuming the original \( 9x^4 - 3x \) might be a typo, or using \( 9x^2 - 3x \) as per the handwritten work)