$j^{3}+3 j^{2} k+3 j k^{2}+k^{3}$ $\newline$ Which of the following is equivalent to the given expression? $\newline$ Choose $1$ answer: $\newline$ (A) $j^{3}+3(j k)^{2}+k^{3}$ $\newline$ (B) $j^{3}+6(j k)^{2}+k^{3}$ $\newline$ (C) $j^{3}+3 j k(j+k)+k^{3}$ $\newline$ (D) $j^{3}+6 j k(j+k)+k^{3}$

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Math Problems

Algebra 1

Compare linear and exponential growth

Full solution

j^{3}+3 j^{2} k+3 j k^{2}+k^{3}

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Which of the following is equivalent to the given expression?

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Choose

1

answer:

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(A)

j^{3}+3(j k)^{2}+k^{3}

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(B)

j^{3}+6(j k)^{2}+k^{3}

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(C)

j^{3}+3 j k(j+k)+k^{3}

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(D)

j^{3}+6 j k(j+k)+k^{3}

Given expression: The given expression is $j^{3}+3j^{2}k+3jk^{2}+k^{3}$ . We need to identify which of the provided choices is equivalent to this expression.
Recognizing the binomial expansion: Recognize that the given expression is the expansion of $(j+k)^3$ based on the binomial theorem, which states that $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ .
Comparing the given expression to the binomial expansion: We can compare the given expression to the binomial expansion: $\newline$ $j^{3}+3j^{2}k+3jk^{2}+k^{3}$ matches the form $a^3 + 3a^2b + 3ab^2 + b^3$ with $a=j$ and $b=k$ .
Comparing the given expression to the answer choices: Now, let's compare the given expression to the answer choices: $\newline$ (A) $j^{3}+3(jk)^{2}+k^{3}$ simplifies to $j^{3}+3j^{2}k^{2}+k^{3}$ , which is not the same as the given expression.
Identifying the correct answer: (B) $j^{3}+6(jk)^{2}+k^{3}$ simplifies to $j^{3}+6j^{2}k^{2}+k^{3}$ , which is also not the same as the given expression.
Identifying the correct answer: (B) $j^{3}+6(jk)^{2}+k^{3}$ simplifies to $j^{3}+6j^{2}k^{2}+k^{3}$ , which is also not the same as the given expression.(C) $j^{3}+3jk(j+k)+k^{3}$ simplifies to $j^{3}+3j^{2}k+3jk^{2}+k^{3}$ , which matches the given expression exactly.
Identifying the correct answer: (B) $j^{3}+6(jk)^{2}+k^{3}$ simplifies to $j^{3}+6j^{2}k^{2}+k^{3}$ , which is also not the same as the given expression.(C) $j^{3}+3jk(j+k)+k^{3}$ simplifies to $j^{3}+3j^{2}k+3jk^{2}+k^{3}$ , which matches the given expression exactly.(D) $j^{3}+6jk(j+k)+k^{3}$ simplifies to $j^{3}+6j^{2}k+6jk^{2}+k^{3}$ , which does not match the given expression.
Identifying the correct answer: (B) $j^{3}+6(jk)^{2}+k^{3}$ simplifies to $j^{3}+6j^{2}k^{2}+k^{3}$ , which is also not the same as the given expression.(C) $j^{3}+3jk(j+k)+k^{3}$ simplifies to $j^{3}+3j^{2}k+3jk^{2}+k^{3}$ , which matches the given expression exactly.(D) $j^{3}+6jk(j+k)+k^{3}$ simplifies to $j^{3}+6j^{2}k+6jk^{2}+k^{3}$ , which does not match the given expression.Therefore, the correct answer is (C) $j^{3}+3jk(j+k)+k^{3}$ , which is equivalent to the given expression.