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Which long division problem can be used to prove the formula for factoring the difference of two perfect cubes?

A. [tex]\[ a - b \longdiv { a ^ { 2 } + a b + b ^ { 2 } } \][/tex]
B. [tex]\[ a + b \longdiv { a ^ { 2 } - a b + b ^ { 2 } } \][/tex]
C. [tex]\[ a + b \sqrt{a^3+0 a^2 b+0 a b^2-b^3} \][/tex]
D. [tex]\[ a - b \longdiv { a ^ { 3 } + 0 a ^ { 2 } b + 0 a b ^ { 2 } - b ^ { 3 } } \][/tex]

Sagot :

GinnyAnswer
To identify which long division problem can be used to prove the formula for factoring the difference of two perfect cubes, we need to recall the formula for factoring such expressions:

The formula for the difference of two cubes is given by:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]

This tells us that the difference of the cubes of two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] can be factored into the product of the difference of the two numbers and a quadratic polynomial involving the two numbers.

Given this, let's analyze the choices provided:

1. [tex]\(a - b \longdiv { a ^ { 2 } + a b + b ^ { 2 } }\)[/tex]

- Here, we are performing division where the divisor is [tex]\(a^2 + ab + b^2\)[/tex]. This polynomial is part of the result we get after factoring, not the entire expression we need to consider for the proof of the factorization.

2. [tex]\(a + b \longdiv { a ^ { 2 } - a b + b ^ { 2 } }\)[/tex]

- Similar to the first option, this division involves a polynomial similar to part of the factorization of a sum of cubes, specifically [tex]\(a^2 - ab + b^2\)[/tex], which isn't applicable for the difference of cubes.

3. [tex]\(a + b \sqrt{a^3+0 a^2 b+0 a b^2-b^3}\)[/tex]

- This notation mixes a radical expression with the sum and difference of terms in a way that isn’t applicable to our straightforward polynomial factorization method.

4. [tex]\(a - b \longdiv { a ^ { 3 } + 0 a ^ { 2 } b + 0 a b ^ { 2 } - b ^ { 3 } }\)[/tex]

- Here, we see that the division corresponds to the expression [tex]\(a^3 - b^3\)[/tex], with [tex]\(a - b\)[/tex] as the divisor. This perfectly aligns with our factorization formula for the difference of cubes:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
- When [tex]\(a^3 - b^3\)[/tex] is divided by [tex]\(a - b\)[/tex], the quotient is indeed [tex]\(a^2 + ab + b^2\)[/tex], thereby verifying the factorization formula.

Therefore, this is the correct option that can be used to prove the formula for factoring the difference of two perfect cubes:

[tex]\[ \boxed{a - b \longdiv { a ^ { 3 } + 0 a ^ { 2 } b + 0 a b ^ { 2 } - b ^ { 3 } }} \][/tex]
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