Answer :
We begin with the polynomial
[tex]$$16x^6 - 9.$$[/tex]
Notice that it can be written as a difference of perfect squares:
[tex]$$16x^6 - 9 = (4x^3)^2 - 3^2.$$[/tex]
Using the difference of squares formula
[tex]$$a^2 - b^2 = (a + b)(a - b),$$[/tex]
with [tex]$a = 4x^3$[/tex] and [tex]$b = 3$[/tex], we factor the expression as
[tex]$$ (4x^3 + 3)(4x^3 - 3). $$[/tex]
In Cecile's work, she rewrites the polynomial as
[tex]$$16x^6 + 12x^3 - 12x^3 - 9,$$[/tex]
which is completely equivalent because the terms [tex]$+12x^3$[/tex] and [tex]$-12x^3$[/tex] cancel each other out. This allows her to use the grouping method (often called the "X method"). She groups the terms as follows:
[tex]$$4x^3(4x^3 + 3) - 3(4x^3 + 3)$$[/tex]
and then factors out the common binomial [tex]$(4x^3 + 3)$[/tex]:
[tex]$$ (4x^3 + 3)(4x^3 - 3). $$[/tex]
Thus, Cecile's factorization is correct.
The final factors of the polynomial are
[tex]$$4x^3 + 3 \quad \text{and} \quad 4x^3 - 3.$$[/tex]
Therefore, the correct answer is that Cecile factored the polynomial correctly.
[tex]$$16x^6 - 9.$$[/tex]
Notice that it can be written as a difference of perfect squares:
[tex]$$16x^6 - 9 = (4x^3)^2 - 3^2.$$[/tex]
Using the difference of squares formula
[tex]$$a^2 - b^2 = (a + b)(a - b),$$[/tex]
with [tex]$a = 4x^3$[/tex] and [tex]$b = 3$[/tex], we factor the expression as
[tex]$$ (4x^3 + 3)(4x^3 - 3). $$[/tex]
In Cecile's work, she rewrites the polynomial as
[tex]$$16x^6 + 12x^3 - 12x^3 - 9,$$[/tex]
which is completely equivalent because the terms [tex]$+12x^3$[/tex] and [tex]$-12x^3$[/tex] cancel each other out. This allows her to use the grouping method (often called the "X method"). She groups the terms as follows:
[tex]$$4x^3(4x^3 + 3) - 3(4x^3 + 3)$$[/tex]
and then factors out the common binomial [tex]$(4x^3 + 3)$[/tex]:
[tex]$$ (4x^3 + 3)(4x^3 - 3). $$[/tex]
Thus, Cecile's factorization is correct.
The final factors of the polynomial are
[tex]$$4x^3 + 3 \quad \text{and} \quad 4x^3 - 3.$$[/tex]
Therefore, the correct answer is that Cecile factored the polynomial correctly.
