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Algebra II (2018 edition)

Unit 4: Lesson 6

Proving polynomial identities

Analyzing polynomial identities
Polynomial identities

Polynomial identities

Felix was asked whether the following equation is an identity:

left parenthesis, x, squared, plus, 1, right parenthesis, squared, equals, left parenthesis, x, squared, minus, 1, right parenthesis, squared, plus, left parenthesis, 2, x, right parenthesis, squared

He performed the following steps:

\begin{aligned} &\phantom{=}(x^2+1)^2 \\\\ \xhookrightarrow{\text{Step }1}\quad&=x^4+x^2+x^2+1 \\\\ \xhookrightarrow{\text{Step }2}\quad&=x^4+2x^2+1 \\\\ \xhookrightarrow{\text{Step }3}\quad&=x^4+2x^2+1-2x^2+2x^2 \\\\ \xhookrightarrow{\text{Step }4}\quad&=(x^4-2x^2+1)+4x^2 \\\\ \xhookrightarrow{\text{Step }5}\quad&=(x^2-1)^2+(2x)^2 \end{aligned}

For this reason, Felix stated that the equation is a true identity.

Is Felix correct? If not, in which step did he make a mistake?

Stuck?

Algebra II (2018 edition)

Unit 4: Lesson 6

Problem