Question

One factor of the function f(x) = x3 − 8x2 + 17x − 10 is (x − 5). Describe how to find the x-intercepts and the y-intercept of the graph of f(x) without using technology. Show your work and include all intercepts in your answer.

Answer 1

the x- intercepts are x = 1, x = 2, x = 5

AND

the y- intercept is y = - 10

Explanation:

given (x - 5) is a factor of f(x) then x = 5 is a zero

dividing f(x) by (x - 5) using synthetic division

5 | 1 - 8 17 - 10

↓ 5 - 15 10

----------------------

1 - 3 2 0 ← remainder = 0 indicates (x - 5) is a factor

quotient = x² - 3x + 2 = (x - 1)(x - 2)

then . .

f(x) = (x - 5)(x - 1)(x - 2) ← in factored form

to find the x- intercepts let f(x) = 0 , th

(x - 5)(x - 1)(x - 2) = 0

equate each factor to zero and solve for x

x - 5 = 0 ⇒ x = 5

x - 1 = 0 ⇒ x = 1

x - 2 = 0 ⇒ x = 2

the x- intercepts are x = 1, x = 2, x = 5

to find the y- intercept let x = 0 , then

f(0) = (0 - 5)(0 - 1)(0 - 2) = (- 5)(- 1)(- 2) = 5 × - 2 = - 10

the y- intercept is y = - 10

Answer 2

see explanation

Explanation:

given (x - 5) is a factor of f(x) then x = 5 is a zero

dividing f(x) by (x - 5) using synthetic division

5 | 1 - 8 17 - 10

↓ 5 - 15 10

----------------------

1 - 3 2 0 ← remainder = 0 indicates (x - 5) is a factor

quotient = x² - 3x + 2 = (x - 1)(x - 2)

then

f(x) = (x - 5)(x - 1)(x - 2) ← in factored form

to find the x- intercepts let f(x) = 0 , that is

(x - 5)(x - 1)(x - 2) = 0

equate each factor to zero and solve for x

x - 5 = 0 ⇒ x = 5

x - 1 = 0 ⇒ x = 1

x - 2 = 0 ⇒ x = 2

the x- intercepts are x = 1, x = 2, x = 5

to find the y- intercept let x = 0 , then

f(0) = (0 - 5)(0 - 1)(0 - 2) = (- 5)(- 1)(- 2) = 5 × - 2 = - 10

the y- intercept is y = - 10