[tex]$80 b^4 - 32 b^2 c^3 + 48 b^4 c$[/tex]

David's work is below:

1. GCF of 80, 32, and 48: 16
2. GCF of [tex]$b^4$[/tex], [tex]$b^2$[/tex], and [tex]$b^4$[/tex]: [tex]$b^2$[/tex]
3. GCF of [tex]$c^3$[/tex] and [tex]$c$[/tex]: [tex]$c$[/tex]
4. GCF of the polynomial: [tex]$16 b^2 c$[/tex]
5. Rewrite as a product of the GCF:
[tex]\[ 16 b^2 c \left( 5 b^2 \right) - 16 b^2 c \left( 2 c^2 \right) + 16 b^2 c \left( 3 b^2 \right) \][/tex]
6. Factor out GCF: [tex]\[ 16 b^2 c \left( 5 b^2 - 2 c^2 + 3 b^2 \right) \][/tex]

Which statements are true about David's work? Check all that apply.

A. The GCF of the coefficients is correct.
B. The GCF of the variable [tex]$b$[/tex] should be [tex]$b^4$[/tex] instead of [tex]$b^2$[/tex].
C. The variable [tex]$c$[/tex] is not common to all terms, so a power of [tex]$c$[/tex] should not have been factored out.
D. The expression in step 5 is equivalent to the given polynomial.
E. In step 6, David applied the distributive property.



Answer :

Let's carefully analyze David's work, step-by-step, and determine the correctness of each statement given below each step.

Step 1:
1. GCD of 80, 32, and 48 is 16.
- This is correct.

Step 2:
2. GCD of \( b^4, b^2 \), and \( b^4 \) is \( b^2 \).
- This is correct since \( b^2 \) is the highest power of \( b \) that is common in all three terms.

Step 3:
3. GCD of \( c^3 \) and \( c \) is \( c \).
- Although this step is correct in isolation, it is irrelevant here because \( c \) does not appear in every term.

Step 4:
4. GCD of the polynomial: \( 16b^2c \).
- This step is incorrect because \( c \) should not be included in the GCD since \( c \) does not appear in every term. The correct GCD should only include \( 16b^2 \).

Step 5:
5. Rewrite as a product of the GCD:
[tex]\[ 16 b^2 c\left(5 b^2\right) - 16 b^2 c\left(2 c^2\right) + 16 b^2 c\left(3 b^2\right) \][/tex]
- This step is incorrect because factoring out \( c \) is not valid as \( c \) is not in every term.

Step 6:
6. Factor out GCD:
[tex]\[ 16 b^2 c\left(5 b^2 - 2 c^2 + 3 b^2\right) \][/tex]
- This step is also incorrect for the same reason as step 5. The term \( c \) should not be factored out from all terms.

### Statements:
1. The GCF of the coefficients is correct.
- True. The GCD of 80, 32, and 48 is indeed 16.

2. The GCF of the variable \( b \) should be \( b^4 \) instead of \( b^2 \).
- False. The GCD of \( b^4, b^2 \), and \( b^4 \) is correctly \( b^2 \).

3. The variable \( c \) is not common to all terms, so any power of \( c \) should not have been factored out.
- True. Since \( c \) is not in every term, factoring it out as part of the GCD is incorrect.

4. The expression in step 5 is equivalent to the given polynomial.
- True. Despite the incorrect factorizations, the rewritten expression in step 5 is algebraically equivalent to the original polynomial.

5. In step 6, David applied the distributive property.
- True. The distributive property is applied in this step despite the incorrect GCD factoring.

Based on this detailed analysis, the correct true statements about David's work are:
- The GCF of the coefficients is correct.
- The variable \( c \) is not common to all terms, so any power of \( c \) should not have been factored out.
- The expression in step 5 is equivalent to the given polynomial.
- In step 6, David applied the distributive property.

Therefore, the true statements from the options given are:

- The GCF of the coefficients is correct.
- The variable c is not common to all terms, so any power of c should not have been factored out.
- The expression in step 5 is equivalent to the given polynomial.
- In step 6, David applied the distributive property.