Question

Given the Function
Find P intercept and x intercept

Answer 1

`\bold{ANSWER:}`
P - intercept is x = 0
X - intercepts is x = -1 and x = -4


`\bold{SOLUTION:}`

To find the P-intercept, we need to find the value of x for which P(x) equals zero. In other words, we need to solve the equation P(x) = 0.

The function P(x) = 4x^4 + 20x^3 + 16x^2 can be factored as follows:

P(x) = 4x^2(x^2 + 5x + 4)

Setting P(x) = 0, we have:

4x^2(x^2 + 5x + 4) = 0

Now we can set each factor equal to zero and solve for x:

4x^2 = 0 --> x = 0

x^2 + 5x + 4 = 0

This quadratic equation can be factored as:

(x + 1)(x + 4) = 0

Setting each factor equal to zero, we have:

x + 1 = 0 --> x = -1

x + 4 = 0 --> x = -4

Therefore, the P-intercept is x = 0, and the x-intercepts are x = -1 and x = -4.

Answer 2

Its P-intercept is (0, 0).

Its x-intercepts are x = -4, x = -1 and x = 0.

Explanation:

The P-intercept is the point where the graph of the function crosses the y-axis, so when x = 0. Therefore, to find the P-intercept, substitute x = 0 into the function:

`\begin{aligned}P(0)&=4(0)^4+20(0)^3+16(0)^2\\&=0+0+0\\&=0\end{aligned}`

Therefore, the P-intercept is at the origin (0, 0).

The x-intercepts are the points at which the graph of the function crosses the x-axis, so when P(x) = 0. Therefore, to find the x-intercepts, we need to set the function to zero and solve for x:

`4x^4+20x^3+16x^2=0`

To solve for x, factor the function.

Factor out the common term 4x²:

`4x^2(x^2+5x+4)=0`

Factor the quadratic:

`4x^2(x^2+x+4x+4)=0`

`4x^2(x(x+1)+4(x+1))=0`

`4x^2(x+1)(x+4)=0`

Set each factor to zero and solve for x:

`4x^2=0 \implies x=0`

`x+1=0 \implies x=-1`

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

Therefore, the x-intercepts are x = -4, x = -1 and x = 0.

Note: As the factor x is squared, indicating a multiplicity of 2, the graph touches or "bounces off" the x-axis at the point (0, 0) rather than crossing the x-axis at this point.