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.
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.
