Answer :
Sure! Let's go through Cecile's work step-by-step to determine whether her factorization of the polynomial [tex]\(16x^6 - 9\)[/tex] is correct or not.
The expression to factor is [tex]\(16x^6 - 9\)[/tex].
### Step 1: Recognize the Difference of Squares
The expression [tex]\(16x^6 - 9\)[/tex] is a difference of squares since:
- [tex]\(16x^6\)[/tex] can be written as [tex]\((4x^3)^2\)[/tex], and
- [tex]\(9\)[/tex] can be written as [tex]\(3^2\)[/tex].
The difference of squares formula is:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
So, we can apply this formula:
- Here, [tex]\(a = 4x^3\)[/tex] and [tex]\(b = 3\)[/tex].
### Step 2: Apply the Difference of Squares Formula
Using the formula, the factorization is:
[tex]\[ (4x^3 + 3)(4x^3 - 3) \][/tex]
Cecile's final factorization step:
[tex]\[ (4x^3 + 3)(4x^3 - 3) \][/tex]
is correct for the difference of squares.
### Step 3: Analyze Intermediate Steps
The concern in Cecile's work arises from the expression:
[tex]\[ 16x^6 + 12x^3 - 12x^3 - 9 \][/tex]
This simplifies to:
[tex]\[ 16x^6 - 9 \][/tex]
since the terms [tex]\(+12x^3\)[/tex] and [tex]\(-12x^3\)[/tex] cancel each other out.
Thus, Cecile's manipulation with [tex]\(16x^6 + 12x^3 - 12x^3 - 9\)[/tex] is indeed equivalent to the original expression [tex]\(16x^6 - 9\)[/tex].
### Conclusion
Cecile's factorization process, using the difference of squares and her intermediate steps, are correct. Therefore, the correct response to the multiple-choice question based on her work is that she factored the polynomial correctly:
[tex]\[ (4x^3 + 3)(4x^3 - 3) \][/tex]
Thus, the answer is Yes, Cecile factored the polynomial correctly.
The expression to factor is [tex]\(16x^6 - 9\)[/tex].
### Step 1: Recognize the Difference of Squares
The expression [tex]\(16x^6 - 9\)[/tex] is a difference of squares since:
- [tex]\(16x^6\)[/tex] can be written as [tex]\((4x^3)^2\)[/tex], and
- [tex]\(9\)[/tex] can be written as [tex]\(3^2\)[/tex].
The difference of squares formula is:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
So, we can apply this formula:
- Here, [tex]\(a = 4x^3\)[/tex] and [tex]\(b = 3\)[/tex].
### Step 2: Apply the Difference of Squares Formula
Using the formula, the factorization is:
[tex]\[ (4x^3 + 3)(4x^3 - 3) \][/tex]
Cecile's final factorization step:
[tex]\[ (4x^3 + 3)(4x^3 - 3) \][/tex]
is correct for the difference of squares.
### Step 3: Analyze Intermediate Steps
The concern in Cecile's work arises from the expression:
[tex]\[ 16x^6 + 12x^3 - 12x^3 - 9 \][/tex]
This simplifies to:
[tex]\[ 16x^6 - 9 \][/tex]
since the terms [tex]\(+12x^3\)[/tex] and [tex]\(-12x^3\)[/tex] cancel each other out.
Thus, Cecile's manipulation with [tex]\(16x^6 + 12x^3 - 12x^3 - 9\)[/tex] is indeed equivalent to the original expression [tex]\(16x^6 - 9\)[/tex].
### Conclusion
Cecile's factorization process, using the difference of squares and her intermediate steps, are correct. Therefore, the correct response to the multiple-choice question based on her work is that she factored the polynomial correctly:
[tex]\[ (4x^3 + 3)(4x^3 - 3) \][/tex]
Thus, the answer is Yes, Cecile factored the polynomial correctly.
