Question

hw problem 1
standard: $p(x) = 2x^4 - 8x^3 + 12x^2 - 8x + 2$
factored: $p(x) = 2(x - 1)^4$
degree:
leading coeff:
end behavior:
y-intercept:
real zeros:
multiplicity
turning points:
workspace:

Explanation:

Step1: Find polynomial degree

The degree is the highest exponent of $x$ in $p(x)=2x^4-8x^3+12x^2-8x+2$.
Degree = $4$

Step2: Identify leading coefficient

The leading coefficient is the coefficient of the highest-degree term.
Leading coefficient = $2$

Step3: Determine end behavior

For even degree and positive leading coefficient: as $x\to+\infty$, $p(x)\to+\infty$; as $x\to-\infty$, $p(x)\to+\infty$.
End behavior: $
earrow/
earrow$

Step4: Calculate y-intercept

Set $x=0$, solve for $p(0)$.
$p(0)=2(0)^4-8(0)^3+12(0)^2-8(0)+2=2$

Step5: Find real zeros

Set factored form $2(x-1)^4=0$, solve for $x$.
$x-1=0 \implies x=1$

Step6: Find zero multiplicity

The factor $(x-1)$ is raised to the 4th power.
Multiplicity = $4$

Step7: Calculate turning points

Maximum turning points = degree $-1$.
Turning points = $4-1=3$ (the graph touches $x=1$ and has a flat point here, so 3 total turning points)

Answer:

Degree: $4$
Leading Coeff: $2$
End Behavior: As $x\to+\infty$, $p(x)\to+\infty$; As $x\to-\infty$, $p(x)\to+\infty$ (or $
earrow/
earrow$)
Y-intercept: $(0, 2)$
Real Zeros: $x=1$
Multiplicity: $4$
Turning Points: $3$