m-demonstrate-the-correct-expansion-and-explain-why-the-students-step-is-invalid

Question

A student claims (x+1)2=$$x2+1(x+1)^2$$ = $$x^2 + 1 by $$distributing the exponent across the sum. Evaluate the claim, demonstrate the correct expansion, and explain why the student's step is invalid.

Accepted Answer

The claim is invalid. Correct expansion: (x+1)2=$$x2+2x+1(x+1)^2$$ = $$x^2 + 2x + 1. $$The student's step fails because exponents do not distribute across sums — cross terms arise from multiplying the binomial by itself.

Answer

The claim is valid for all xx because squaring distributes over addition, so $$x2+1x^2 + 1 is $$correct; cross terms can be ignored for simplicity.

Answer

The claim is valid only when x=0x = 0 or x=−1x = -1, otherwise the missing 2x2x term makes the equality false in general.

Answer

The claim is invalid; the correct expansion is $$x2+12x+2xx^2$$ + \frac{1}{2x} + 2x when distributing an exponent through a sum.

A student claims (x+1)2=x2+1(x+1)^2 = x^2 + 1 by distributing the exponent across the sum. Evaluate the claim, demonstrate the correct expansion, and explain why the student's step is invalid.

A student claims $(x + 1)^{2} = x^{2} + 1$ by distributing the exponent across the sum. Evaluate the claim, demonstrate the correct expansion, and explain why the student's step is invalid.