Question

NO LINKS!! Please help me with this polynomial function activity 6. Also make a graph.​

Answer 1

x-intercepts: -4, -2, 1 and 3

y-intercept: 24

`\begin{array}c\cline{1-6}x&-5&-3&0&2&4\\\cline{1-6}y&144&-24&24&-24&144\\\cline{1-6}\end{array}`

Explanation:

Given polynomial:

`y=(x+4)(x+2)(x-1)(x-3)`

The x-intercepts are the points at which the curve intersects the x-axis, so when the function equals zero.

Zero Product Property

If a ⋅ b = 0 then either a = 0 or b = 0 (or both).

Therefore, to find the x-intercepts, set each factor of the given polynomial equal to zero and solve for x:

Therefore:

`\implies (x+4)=0 \implies x=-4`

`\implies (x+2)=0 \implies x=-2`

`\implies (x-1)=0 \implies x=1`

`\implies (x-3)=0 \implies x=3`

Therefore, the x-intercepts are -4, -2, 1 and 3.

The y-intercept is the point at which the curve intersects the y-axis, so when x is zero.

To find the the y-intercept, substitute x = 0 into the given polynomial:

`\implies y=(0+4)(0+2)(0-1)(0-3)`

`\implies y=(4)(2)(-1)(-3)`

`\implies y=(8)(-1)(-3)`

`\implies y=(-8)(-3)`

`\implies y=24`

Therefore, the y-intercept is 24.

To find the other points on the graph, substitute each value of x into the polynomial and solve for y:

`\begin{aligned}x=-5 \implies y&=(-5+4)(-5+2)(-5-1)(-5-3)\\&=(-1)(-3)(-6)(-8)\\&=(3)(-6)(-8)\\&=(-18)(-8)\\&=144\end{aligned}`

`\begin{aligned}x=-3 \implies y&=(-3+4)(-3+2)(-3-1)(-3-3)\\&=(1)(-1)(-4)(-6)\\&=(-1)(-4)(-6)\\&=(4)(-6)\\&=-24\end{aligned}`

`\begin{aligned}x=2 \implies y&=(2+4)(2+2)(2-1)(2-3)\\&=(6)(4)(1)(-1)\\&=(24)(1)(-1)\\&=(24)(-1)\\&=-24\end{aligned}`

`\begin{aligned}x=4 \implies y&=(4+4)(4+2)(4-1)(4-3)\\&=(8)(6)(3)(1)\\&=(48)(3)(1)\\&=(144)(1)\\&=144\end{aligned}`

Therefore:

`\large\begin{array}c\cline{1-6}x&-5&-3&0&2&4\\\cline{1-6}y&144&-24&24&-24&144\\\cline{1-6}\end{array}`