Respuesta :
Step-by-step explanation:
x^3 - 7x^2 - 14x - 8 in factored form is
equal to (x-1)(x-2)(x-4).
Solving for x-intercepts:
We are actually able to solve for all x-intercepts without the given factor. But since we are given one of the factors, our job becomes much easier.
Using synthetic division, or long division, we factor out the x intercept 4. Which leaves us with thepolynomial x^2 - 3x + 2.
From here we can separate the polynomial into two binomials.
x^2 - 3x + 2 = (x-1)(x-2). Giving us all 3 x-intercepts.
Using Descartes' rules we can identify before even starting the problem how many real x-intercepts there are (Not needed for this problem).
Solving for y- intercepts:
The y-intercept is always the coefficient that does not have any assigned x-variables.
The coefficient is -8, thus the y intercept.
If unsure of the y-intercept, you can always plug in x = 0. Solving for the y-intercept will give you the value of f(0).
If there is no coefficient, the y intercept is equal to zero.
Answer: x-intercepts = 1,2, and 4, y-intercept = -8
[tex]\large{\underline{{\red{F}{\pink{in}{\color{blue}{din}{\color{gold}{g}{\color{aqua}{\:}{\color{lime}{x-intercept}}}}}}}}}[/tex]
[tex]f(x) = x ^{3} - 7x ^{2} + 14x - 8[/tex]
- To find x-intercept/zero, substitute f(x)=0
[tex]0 = x ^{3} - 7x ^{2} + 14x - 8[/tex]
- Move the expression to the left-hand side and change its sign
[tex]0 - x ^{3} + 7x ^{2} - 14x + 8 = 0[/tex]
- When adding or subtracting 0, the quantity does not change
[tex] - x ^{3} + 7x ^{2} - 14x + 8 = 0[/tex]
- Factor out the negative sign from the expression
[tex] - (x ^{3} - 8) + 7x ^{2} - 14x = 0[/tex]
- Factor out 7x from the expression
[tex] - (x ^{3} - 8) + 7x \times (x - 2) = 0[/tex]
- Use a^3-b^3=(a-b)(a^2+ab+b^2) to factor the expression
[tex] - (x - 2) \times (x ^{2} + 2x + 4) + 7x \times (x - 2) = 0[/tex]
- Factor out -(x-2) from the expression
[tex] - (x - 2) \times (x ^{2} + 2x + 4 - 7x) = 0[/tex]
- Collect like terms
[tex] - (x - 2) \times (x ^{2} - 5x + 4) = 0[/tex]
- Write -5x as difference
[tex] - (x - 2) \times (x ^{2} - x - 4x + 4) = 0[/tex]
- Factor out x and -4 from the expression
[tex] - (x - 2) \times (x \times (x - 1) - 4(x - 1)) = 0[/tex]
- Factor out x-1 from the expression
[tex] - (x - 2) \times (x - 1) \times (x - 4) = 0[/tex]
- Change the signs on both sides of the equation
[tex](x - 2)(x - 1)(x - 4) = 0[/tex]
- When the product of factors equals 0, at least one factor is 0
[tex]x - 2 = 0 \\ x - 4 = 0 \\ x - 1 = 0[/tex]
- Solve the equation for x
[tex]x = 2 \\ x = 4 \\ x = 1[/tex]
- The equation has 3 solutions
[tex]x_{1} =2 , x_{2} = 4 , x_{3} = 1[/tex]
[tex]\rule{300pt}{3pt}[/tex]
[tex]\large{\underline{{\red{F}{\pink{in}{\color{blue}{din}{\color{gold}{g}{\color{aqua}{\:}{\color{lime}{y-intercept}}}}}}}}}[/tex]
- To find y-intercept, substitute x=0
[tex]f(0) = 0 ^{3} - 7 \times 0 ^{2} + 14 \times 0 - 8[/tex]
- 0 raised to any power equals 0
- 0 multiplied by any number equals 0
[tex]f(0) = 0 - 7 \times 0 - 8[/tex]
- When adding or subtracting 0, the quantity does not change
[tex]f(0) = - 0 - 8[/tex]
[tex]f(0) = - 8[/tex]
