Question

hw problem 2
standard: $q(x) = 2x^3 + 6x^2 - 8$
factored: $q(x) = 2(x - 1)(x + 2)^2$
degree:
leading coeff:
end behavior:
y-intercept:
real zeros:
multiplicity
turning points:
workspace:

hw problem 3
standard: $r(x) = -2x^4 - 4x^3 + 4x + 2$
factored: $r(x) = -2(x + 1)^3(x - 1)$
degree:
leading coeff:
end behavior:
y-intercept:
real zeros:
multiplicity
turning points:
workspace:

Explanation:

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HW Problem 2

Step1: Identify polynomial degree

The highest power of $x$ in $q(x)=2x^3+6x^2-8$ is 3.

Step2: Identify leading coefficient

The coefficient of the highest-degree term $2x^3$ is 2.

Step3: Determine end behavior

For odd degree, positive leading coefficient: as $x\to+\infty$, $q(x)\to+\infty$; as $x\to-\infty$, $q(x)\to-\infty$.

Step4: Calculate y-intercept

Set $x=0$: $q(0)=2(0)^3+6(0)^2-8=-8$.

Step5: Find real zeros

From factored form $q(x)=2(x-1)(x+2)^2$, set to 0: $x-1=0$ or $x+2=0$.

Step6: Identify zero multiplicity

$x=1$: exponent 1 (multiplicity 1); $x=-2$: exponent 2 (multiplicity 2).

Step7: Calculate turning points

For degree $n$, max turning points = $n-1=3-1=2$.

Step1: Identify polynomial degree

The highest power of $x$ in $r(x)=-2x^4-4x^3+4x+2$ is 4.

Step2: Identify leading coefficient

The coefficient of the highest-degree term $-2x^4$ is $-2$.

Step3: Determine end behavior

For even degree, negative leading coefficient: as $x\to+\infty$, $r(x)\to-\infty$; as $x\to-\infty$, $r(x)\to-\infty$.

Step4: Calculate y-intercept

Set $x=0$: $r(0)=-2(0)^4-4(0)^3+4(0)+2=2$.

Step5: Find real zeros

From factored form $r(x)=-2(x+1)^3(x-1)$, set to 0: $x+1=0$ or $x-1=0$.

Step6: Identify zero multiplicity

$x=-1$: exponent 3 (multiplicity 3); $x=1$: exponent 1 (multiplicity 1).

Step7: Calculate turning points

For degree $n$, max turning points = $n-1=4-1=3$.

Answer:

• Degree: $3$
• Leading Coeff: $2$
• End Behavior: As $x\to+\infty$, $q(x)\to+\infty$; As $x\to-\infty$, $q(x)\to-\infty$
• Y-intercept: $(0, -8)$
• Real Zeros: $x=1$, $x=-2$
• Multiplicity: $x=1$ (multiplicity 1), $x=-2$ (multiplicity 2)
• Turning Points: $2$

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HW Problem 3