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Answer :
To solve the problem, we need to find two things for the function [tex]\( f(x) = 2x^4 + 4x^3 - 70x^2 \)[/tex]:
1. The f-intercept: This is the value of the function when [tex]\( x = 0 \)[/tex].
2. The x-intercept(s): These are the points where the function equals zero, i.e., the solutions to [tex]\( f(x) = 0 \)[/tex].
### Step 1: Finding the f-intercept
The f-intercept is the value of the function [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = 2(0)^4 + 4(0)^3 - 70(0)^2 = 0
\][/tex]
So, the f-intercept is [tex]\( 0 \)[/tex].
### Step 2: Finding the x-intercepts
The x-intercepts are the solutions to the equation [tex]\( f(x) = 0 \)[/tex].
- Set the function equal to zero:
[tex]\[
2x^4 + 4x^3 - 70x^2 = 0
\][/tex]
- Factor out the greatest common factor:
[tex]\[
x^2(2x^2 + 4x - 70) = 0
\][/tex]
- This gives you one solution: [tex]\( x = 0 \)[/tex].
- Now solve the quadratic equation [tex]\( 2x^2 + 4x - 70 = 0 \)[/tex].
- You can factor the quadratic or use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]. In factored form, the solutions are:
[tex]\[
(x + 7)(x - 5) = 0
\][/tex]
- Therefore, the additional solutions are [tex]\( x = -7 \)[/tex] and [tex]\( x = 5 \)[/tex].
Thus, the x-intercepts are [tex]\( x = -7 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 5 \)[/tex].
### Final Answer:
- The f-intercept is [tex]\( 0 \)[/tex].
- The x-intercepts are [tex]\( x = -7 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 5 \)[/tex].
1. The f-intercept: This is the value of the function when [tex]\( x = 0 \)[/tex].
2. The x-intercept(s): These are the points where the function equals zero, i.e., the solutions to [tex]\( f(x) = 0 \)[/tex].
### Step 1: Finding the f-intercept
The f-intercept is the value of the function [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = 2(0)^4 + 4(0)^3 - 70(0)^2 = 0
\][/tex]
So, the f-intercept is [tex]\( 0 \)[/tex].
### Step 2: Finding the x-intercepts
The x-intercepts are the solutions to the equation [tex]\( f(x) = 0 \)[/tex].
- Set the function equal to zero:
[tex]\[
2x^4 + 4x^3 - 70x^2 = 0
\][/tex]
- Factor out the greatest common factor:
[tex]\[
x^2(2x^2 + 4x - 70) = 0
\][/tex]
- This gives you one solution: [tex]\( x = 0 \)[/tex].
- Now solve the quadratic equation [tex]\( 2x^2 + 4x - 70 = 0 \)[/tex].
- You can factor the quadratic or use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]. In factored form, the solutions are:
[tex]\[
(x + 7)(x - 5) = 0
\][/tex]
- Therefore, the additional solutions are [tex]\( x = -7 \)[/tex] and [tex]\( x = 5 \)[/tex].
Thus, the x-intercepts are [tex]\( x = -7 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 5 \)[/tex].
### Final Answer:
- The f-intercept is [tex]\( 0 \)[/tex].
- The x-intercepts are [tex]\( x = -7 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 5 \)[/tex].
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