Unit 5 Review

Parity of $x^6$.

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Domain of $x^-3$.

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Range of $\frac{1}{x}^2$.

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Faster for large x, $x^3$ or $x^5$?

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Degree of $5x^3-2x+8$.

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Leading term of $-4x^7+x^9-1$.

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Is $sqrt(x)+x$ polynomial?

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Explain why this conclusion is valid: Standard form $-2x^4+5x^2+7$; degree $4$; leading coefficient $-2$. Use the foundation problem as evidence: For $7-2x^4+5x^2$.

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

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End behavior of $-x^4+2$.

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Can odd degree have same ends?

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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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Zeros of $(x-5)(x+1)$.

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Factor for zero$$7$$

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Can complex zero be x-intercept?

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Explain why this conclusion is valid: $x=-2,0,2$. Use the foundation problem as evidence: Zeros of $x^3-4x$.

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

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Does even multiplicity change sign?

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Degree of $(x+1)^2(x-3)^4$.

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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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Sign for $x<-1$.

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Max turns degree $5$.

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Why no holes?

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Explain why this conclusion is valid: Left down, right up; cross at $-2,$ touch at $1$; y-intercept $2$. Use the foundation problem as evidence: Sketch $(x+2)(x-1)^2$.

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Max turns degree $7$.

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Does every zero create a turn?

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Which behavior guarantees absolute maximum?

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Show $x^3-2$ root in $(1,2)$.

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Can double root show no sign change?

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Why continuity matters?

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Explain why this conclusion is valid: $P(1)=-1$ and $P(2)=5,$ so a root lies in $(1,2)$. Use the foundation problem as evidence: For $x^3-x-1$ on $(1,2)$.

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Divide $x^2-9$ by $x-3$.

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$c=-3$ corresponds to divisor.

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Remainder degree condition.

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Explain why this conclusion is valid: Quotient $x^2-x-2,$ remainder $0$. Use the foundation problem as evidence: Divide $x^3+2x^2-5x-6$ by $x+3$.

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Remainder of $x^2+3x+1$ by $x-2$.

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If $P(4)=7,$ remainder by $x-4$.

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Meaning repeated division.

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Explain why this conclusion is valid: Yes, because $P(-2)=0$. Use the foundation problem as evidence: Is $x+2$ a factor of $x^3-3x+2$?

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Candidates for $x^3+2x-8$.

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Does theorem guarantee rational root?

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After confirming root.

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Explain why this conclusion is valid: $\pm 1,\pm 2,\pm 4,\frac{\pm 1}{2}$. Use the foundation problem as evidence: Candidates for $2x^3+x^2-8x-4$.

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Zeros count degree $7$.

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Factor for zeros$$\pm 2i$$

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Can nonreal roots repeat?

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Explain why this conclusion is valid: Real factor $x^2-6x+13$. Use the foundation problem as evidence: Zeros $3\pm 2i$.

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

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Why keep factored form?

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Why can bisection miss even root?

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