Unit 5 Flexible Practice

Parity of $x^6$.

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Explain why this conclusion is valid: Even, all-real domain, both ends down, range $(-∞,0]$. Use the foundation problem as evidence: Describe $-2x^4$.

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Create a nearby example by changing one number or condition in this prompt: Range of $\frac{1}{x}^2$. Predict the effect, solve your new example, and compare it with the original.

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Is $sqrt(x)+x$ polynomial?

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Correct this reasoning and identify the first unsafe assumption: A frequent error is replacing the polynomial by its leading term near the origin.

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End behavior of $x^8-3x$.

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Explain why this conclusion is valid: Left end rises and right end falls. Use the foundation problem as evidence: Determine end behavior of $-3x^5+x^2$.

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Create a nearby example by changing one number or condition in this prompt: Can even degree have opposite ends? Predict the effect, solve your new example, and compare it with the original.

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Intercept for zero $-3$.

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Correct this reasoning and identify the first unsafe assumption: A frequent error is reporting an intercept point as a zero or treating a decimal as exact.

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Multiplicity of zero $4$ in $(x-4)^3$.

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Explain why this conclusion is valid: $-2$ touches; $1$ crosses with flattening. Use the foundation problem as evidence: Analyze $(x+2)^2(x-1)^3$.

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Create a nearby example by changing one number or condition in this prompt: Does even multiplicity change sign? Predict the effect, solve your new example, and compare it with the original.

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Max turns degree $5$.

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Correct this reasoning and identify the first unsafe assumption: A frequent error is adding decorative wiggles unsupported by the algebra.

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Max turns degree $2$.

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Solve the representation example, then name the feature of turning points and local versus global behavior that it illustrates: Max turns degree$$7$$

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Write a short verification checklist for turning points and local versus global behavior, then apply it to one worked example from this lesson.

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Advantage of bisection.

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Connect two representations for this example: For $x^3-x-1$ on $(1,2)$. Describe what a graph, table, mapping, or algebraic form would have to show.

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$c=-3$ corresponds to divisor.

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Solve the representation example, then name the feature of polynomial division that it illustrates: $c=-3$ corresponds to divisor.

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Write a short verification checklist for polynomial division, then apply it to one worked example from this lesson.

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Meaning repeated division.

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Connect two representations for this example: Is $x+2$ a factor of $x^3-3x+2$? Describe what a graph, table, mapping, or algebraic form would have to show.

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If constant term zero.

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Solve the representation example, then name the feature of rational-root candidates and root search that it illustrates: If constant term zero.

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Write a short verification checklist for rational-root candidates and root search, then apply it to one worked example from this lesson.

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Can nonreal roots repeat?

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Connect two representations for this example: Zeros $3\pm 2i$. Describe what a graph, table, mapping, or algebraic form would have to show.

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Real roots $2$ and $1\pm i$.

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Solve the representation example, then name the feature of building polynomial models and finding numerical roots that it illustrates: Real roots $2$ and $1\pm i$.

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