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Answer :
To determine which of the given expressions is a factor of [tex]\(x^3 - 64\)[/tex], we need to factorize the polynomial [tex]\(x^3 - 64\)[/tex]. The expression [tex]\(x^3 - 64\)[/tex] is a difference of cubes since [tex]\(64\)[/tex] can be written as [tex]\(4^3\)[/tex].
The formula for the difference of cubes is:
[tex]\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\][/tex]
In this case, [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]. Applying the formula:
[tex]\[x^3 - 64 = x^3 - 4^3 = (x - 4)(x^2 + 4x + 16)\][/tex]
So, the factors of [tex]\(x^3 - 64\)[/tex] are [tex]\((x - 4)\)[/tex] and [tex]\((x^2 + 4x + 16)\)[/tex].
Let's match these factors with the options provided:
- A. [tex]\(x - 4\)[/tex]
- B. [tex]\(x + 4\)[/tex]
- C. [tex]\(x + 64\)[/tex]
- D. [tex]\(x^2 + 16\)[/tex]
- E. [tex]\(x^2 - 4x + 16\)[/tex]
From our factorization, we can see that:
- [tex]\( (x - 4) \)[/tex] is a factor.
- [tex]\( (x^2 + 4x + 16) \)[/tex] is a factor.
Therefore, the correct matching option from the list is:
A. [tex]\(x - 4\)[/tex]
And so, the answer is [tex]\(A)\ x-4\)[/tex].
The formula for the difference of cubes is:
[tex]\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\][/tex]
In this case, [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]. Applying the formula:
[tex]\[x^3 - 64 = x^3 - 4^3 = (x - 4)(x^2 + 4x + 16)\][/tex]
So, the factors of [tex]\(x^3 - 64\)[/tex] are [tex]\((x - 4)\)[/tex] and [tex]\((x^2 + 4x + 16)\)[/tex].
Let's match these factors with the options provided:
- A. [tex]\(x - 4\)[/tex]
- B. [tex]\(x + 4\)[/tex]
- C. [tex]\(x + 64\)[/tex]
- D. [tex]\(x^2 + 16\)[/tex]
- E. [tex]\(x^2 - 4x + 16\)[/tex]
From our factorization, we can see that:
- [tex]\( (x - 4) \)[/tex] is a factor.
- [tex]\( (x^2 + 4x + 16) \)[/tex] is a factor.
Therefore, the correct matching option from the list is:
A. [tex]\(x - 4\)[/tex]
And so, the answer is [tex]\(A)\ x-4\)[/tex].
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