Answer:
• Degree: 3
,• Graph below
Explanation:
Given the function:
[tex]P\left(x\right)=\left(x+10\right)\left(x+7\right)\left(x-12\right)[/tex]• The roots of the function are -10, -7, and 12.
,• The function is of degree 3.
When x=0
[tex]\begin{gathered} P(x)=(x+10)(x+7)(x-12) \\ P(0)=(0+10)(0+7)(0-12)=-840 \end{gathered}[/tex]The y-intercept is at (0, -840).
Now, to graph the polynomial, first, find we determine the critical points by finding the derivative of the function.
First, expand P(x)
[tex]\begin{gathered} P(x)=(x+10)(x^2-12x+7x-84) \\ =(x+10)(x^2-5x-84) \\ =x^3-5x^2-84x+10x^2-50x-840 \\ P(x)=x^3+5x^2-134x-840 \end{gathered}[/tex]Next, find the derivative:
[tex]P^{\prime}(x)=3x^2+10x-134[/tex]Set the derivative equal to 0 and solve for x:
[tex]\begin{gathered} 3x^2+10x-134=0 \\ \implies x=-8.55,5.22 \end{gathered}[/tex]Find the corresponding y values at x=-8.55 and 5.22.
[tex]\begin{gathered} P(x)=(x+10)(x+7)(x-12) \\ P(-8.55)=(-8.55+10)(-8.55+7)(-8.55-12)=46.19 \\ P(5.22)=(5.22+10)(5.22+7)(5.22-12)=-1261.00 \end{gathered}[/tex]So far, we have the following:
• Roots: (-10, 0), (-7, 0), (12, 0)
,• The y-intercept is at (0,-840).
,• Critical Points: (-8.55, 46.19), (5.22, -1261)
A sketch of these points is given below:
You can use this as a better guide:
