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題目

解題步驟

1. Understand the question

   determine the zeros and the vertical intercept of the polynomial function:
   $f(x)=2(x-8)^2(x+6)(x-11)$
   the zero(s) of the is/are
   the horizontal intercept(s) is/are
   the vertical intercept is
   the degree of the polynomial function is

2. Explanation

   Step1: Find zeros via factor roots

   Set each factor equal to 0:
   $x-8=0 \implies x=8$ (multiplicity 2)
   $x+6=0 \implies x=-6$
   $x-11=0 \implies x=11$

   Step2: Identify horizontal intercepts

   Horizontal intercepts are the zero pairs $(x,0)$:
   $(8,0), (-6,0), (11,0)$

   Step3: Calculate vertical intercept

   Set $x=0$ and solve for $f(0)$:
   $$
   \begin{align*}
   f(0)&=2(0-8)^2(0+6)(0-11)\\
   &=2(64)(6)(-11)\\
   &=2\times64\times(-66)\\
   &=128\times(-66)\\
   &=-8448
   \end{align*}
   $$

   Step4: Find polynomial degree

   Sum exponents of factors: $2+1+1=4$

3. Final answer

   The zero(s) of the function is/are $x=8$ (multiplicity 2), $x=-6$, $x=11$
   The horizontal intercept(s) is/are $(8, 0)$, $(-6, 0)$, $(11, 0)$
   The vertical intercept is $(0, -8448)$
   The degree of the polynomial function is $4$

答案

Question Analysis

OCR Text

determine the zeros and the vertical intercept of the polynomial function:
$f(x)=2(x-8)^2(x+6)(x-11)$
the zero(s) of the is/are
the horizontal intercept(s) is/are
the vertical intercept is
the degree of the polynomial function is

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