factor each completely.
1) $x^2 - 5x$
2) $2b^3 + 24b^2 + 70b$
3) $k^2 - 2k - 80$
4) $6x^3 + 30x^2 - 216x$
5) $a^3 - 8a^2$
6) $x^2 + 7x - 8$
7) $x^2 + 4x - 32$
8) $p^3 - 81p$
date

Question

Accepted Answer

### Problem 1: $ x^2 - 5x $
# Explanation:
## Step1: Identify common factor
Both terms have $ x $ as a common factor.
$ x^2 - 5x = x(x - 5) $
# Answer:
$ x(x - 5) $

### Problem 2: $ 2b^3 + 24b^2 + 70b $
# Explanation:
## Step1: Factor out GCF
The greatest common factor (GCF) of $ 2b^3 $, $ 24b^2 $, and $ 70b $ is $ 2b $.
$ 2b^3 + 24b^2 + 70b = 2b(b^2 + 12b + 35) $
## Step2: Factor quadratic
Factor $ b^2 + 12b + 35 $. We need two numbers that multiply to 35 and add to 12. Those numbers are 5 and 7.
$ b^2 + 12b + 35 = (b + 5)(b + 7) $
## Step3: Combine factors
$ 2b(b + 5)(b + 7) $
# Answer:
$ 2b(b + 5)(b + 7) $

### Problem 3: $ k^2 - 2k - 80 $
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to -80 and add to -2. The numbers are -10 and 8.
## Step2: Factor the quadratic
$ k^2 - 2k - 80 = (k - 10)(k + 8) $
# Answer:
$ (k - 10)(k + 8) $

### Problem 4: $ 6x^3 + 30x^2 - 216x $
# Explanation:
## Step1: Factor out GCF
The GCF of $ 6x^3 $, $ 30x^2 $, and $ -216x $ is $ 6x $.
$ 6x^3 + 30x^2 - 216x = 6x(x^2 + 5x - 36) $
## Step2: Factor quadratic
Factor $ x^2 + 5x - 36 $. We need two numbers that multiply to -36 and add to 5. Those numbers are 9 and -4.
$ x^2 + 5x - 36 = (x + 9)(x - 4) $
## Step3: Combine factors
$ 6x(x + 9)(x - 4) $
# Answer:
$ 6x(x + 9)(x - 4) $

### Problem 5: $ a^3 - 8a^2 $
# Explanation:
## Step1: Identify common factor
Both terms have $ a^2 $ as a common factor.
$ a^3 - 8a^2 = a^2(a - 8) $
# Answer:
$ a^2(a - 8) $

### Problem 6: $ x^2 + 7x - 8 $
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to -8 and add to 7. The numbers are 8 and -1.
## Step2: Factor the quadratic
$ x^2 + 7x - 8 = (x + 8)(x - 1) $
# Answer:
$ (x + 8)(x - 1) $

### Problem 7: $ x^2 + 4x - 32 $
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to -32 and add to 4. The numbers are 8 and -4.
## Step2: Factor the quadratic
$ x^2 + 4x - 32 = (x + 8)(x - 4) $
# Answer:
$ (x + 8)(x - 4) $

### Problem 8: $ p^3 - 81p $
# Explanation:
## Step1: Factor out GCF
The GCF of $ p^3 $ and $ -81p $ is $ p $.
$ p^3 - 81p = p(p^2 - 81) $
## Step2: Factor difference of squares
$ p^2 - 81 $ is a difference of squares: $ p^2 - 9^2 = (p - 9)(p + 9) $
## Step3: Combine factors
$ p(p - 9)(p + 9) $
# Answer:
$ p(p - 9)(p + 9) $

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( x^2 - 5x \)

Step1: Identify common factor

Step1: Factor out GCF

Step2: Factor quadratic

Step3: Combine factors

Step1: Find two numbers

Step2: Factor the quadratic

Answer:

Problem 2: \( 2b^3 + 24b^2 + 70b \)