Question

One factor of the function f(x) = x ^ 3 - 8x ^ 2 + 17x - 10.5(x - 5) Describe how to find the x-intercepts and the y-intercept of the graph of f(x) without using technologyShow your work and include all intercepts in your answer.

Answer 1

x-intercept(s): (−1.9900717, 0)

y-intercept(s): (0, 52.5)

Explanation:

To find the x-intercept, substitute in
`0` for
`y` and solve for
`x`. To find the y-intercept, substitute in
`0` for
`x` and solve for
`y` .

`f(x)=y`

`(x)=x`

x-intercept

To find the x-intercept(s), substitute in
`0` for
`y` and solve for
`x` .

`f(x) = x ^ 3 - 8x ^ 2 + 17x - 10.5(x - 5`

`=\bold{0}=x^3-8x^2+17x-10.5\left(x-5\right)`

`=0\cdot \:10=x^3\cdot \:10-8x^2\cdot \:10+17x\cdot \:10-10.5\left(x-5\right)\cdot \:10`

`=0=10x^3-80x^2+170x-105\left(x-5\right)`

`=0=10x^3-80x^2+65x+525`

`=10x^3-80x^2+65x+525=0`

`=10x^2-99.90071\dots x+263.80959\dots \approx \:0`

`\bold{x\approx \:-1.99007\dots}`

y-intercept

`f(x) = x ^ 3 - 8x ^ 2 + 17x - 10.5(x - 5`

`=y=\bold{0}^3-8(\bold{0})^2+17(\bold{0})-10.5\left(\bold{0}-5\right)`

`=y=0-8\cdot \:0+17\left(0\right)-10.5\left(0-5\right)`

`=y=0-8\cdot \:0+17\cdot \:0-10.5\left(0-5\right)`

`=y=0-0+0-\left(-52.5\right)`

`=0-0+0+52.5`

`\bold{y=52.5}`