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Sagot :
To determine the intercepts of the function [tex]\( P(r) = 3r^4 + 9r^3 - 30r^2 \)[/tex], let's proceed step by step.
### Finding the [tex]\( P \)[/tex]-intercept:
The [tex]\( P \)[/tex]-intercept occurs where the function intersects the [tex]\( P \)[/tex]-axis. This happens when [tex]\( r = 0 \)[/tex].
1. Substitute [tex]\( r = 0 \)[/tex] into the function:
[tex]\[ P(0) = 3(0)^4 + 9(0)^3 - 30(0)^2 = 0 \][/tex]
Hence, the [tex]\( P \)[/tex]-intercept is:
[tex]\[ (0, 0) \][/tex]
### Finding the [tex]\( r \)[/tex]-intercepts:
The [tex]\( r \)[/tex]-intercepts occur where the function intersects the [tex]\( r \)[/tex]-axis. This happens when [tex]\( P(r) = 0 \)[/tex]. We need to solve the equation:
[tex]\[ 3r^4 + 9r^3 - 30r^2 = 0 \][/tex]
Step-by-Step Solution:
1. Factor out the greatest common factor, which is [tex]\( 3r^2 \)[/tex]:
[tex]\[ 3r^2(r^2 + 3r - 10) = 0 \][/tex]
2. Solve for [tex]\( r \)[/tex] by setting each factor equal to zero:
[tex]\[ 3r^2 = 0 \quad \text{or} \quad r^2 + 3r - 10 = 0 \][/tex]
3. Solving [tex]\( 3r^2 = 0 \)[/tex]:
[tex]\[ r^2 = 0 \implies r = 0 \][/tex]
4. Solving the quadratic equation [tex]\( r^2 + 3r - 10 = 0 \)[/tex] using the quadratic formula [tex]\( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ a = 1, \quad b = 3, \quad c = -10 \][/tex]
[tex]\[ r = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1} \][/tex]
[tex]\[ r = \frac{-3 \pm \sqrt{9 + 40}}{2} \][/tex]
[tex]\[ r = \frac{-3 \pm \sqrt{49}}{2} \][/tex]
[tex]\[ r = \frac{-3 \pm 7}{2} \][/tex]
This gives us two solutions:
[tex]\[ r = \frac{-3 + 7}{2} = 2 \quad \text{and} \quad r = \frac{-3 - 7}{2} = -5 \][/tex]
Thus, the [tex]\( r \)[/tex]-intercepts are:
[tex]\[ (-5, 0), (0, 0), (2, 0) \][/tex]
### Summary:
- The [tex]\( P \)[/tex]-intercept is:
[tex]\[ (0, 0) \][/tex]
- The [tex]\( r \)[/tex]-intercepts are:
[tex]\[ (-5, 0), (0, 0), (2, 0) \][/tex]
Therefore, the solution to the given question is:
Its [tex]\( P \)[/tex]-intercept is [tex]\((0, 0)\)[/tex]
Its [tex]\( r \)[/tex]-intercept(s) is/are [tex]\((-5, 0), (0, 0), (2, 0)\)[/tex]
### Finding the [tex]\( P \)[/tex]-intercept:
The [tex]\( P \)[/tex]-intercept occurs where the function intersects the [tex]\( P \)[/tex]-axis. This happens when [tex]\( r = 0 \)[/tex].
1. Substitute [tex]\( r = 0 \)[/tex] into the function:
[tex]\[ P(0) = 3(0)^4 + 9(0)^3 - 30(0)^2 = 0 \][/tex]
Hence, the [tex]\( P \)[/tex]-intercept is:
[tex]\[ (0, 0) \][/tex]
### Finding the [tex]\( r \)[/tex]-intercepts:
The [tex]\( r \)[/tex]-intercepts occur where the function intersects the [tex]\( r \)[/tex]-axis. This happens when [tex]\( P(r) = 0 \)[/tex]. We need to solve the equation:
[tex]\[ 3r^4 + 9r^3 - 30r^2 = 0 \][/tex]
Step-by-Step Solution:
1. Factor out the greatest common factor, which is [tex]\( 3r^2 \)[/tex]:
[tex]\[ 3r^2(r^2 + 3r - 10) = 0 \][/tex]
2. Solve for [tex]\( r \)[/tex] by setting each factor equal to zero:
[tex]\[ 3r^2 = 0 \quad \text{or} \quad r^2 + 3r - 10 = 0 \][/tex]
3. Solving [tex]\( 3r^2 = 0 \)[/tex]:
[tex]\[ r^2 = 0 \implies r = 0 \][/tex]
4. Solving the quadratic equation [tex]\( r^2 + 3r - 10 = 0 \)[/tex] using the quadratic formula [tex]\( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ a = 1, \quad b = 3, \quad c = -10 \][/tex]
[tex]\[ r = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1} \][/tex]
[tex]\[ r = \frac{-3 \pm \sqrt{9 + 40}}{2} \][/tex]
[tex]\[ r = \frac{-3 \pm \sqrt{49}}{2} \][/tex]
[tex]\[ r = \frac{-3 \pm 7}{2} \][/tex]
This gives us two solutions:
[tex]\[ r = \frac{-3 + 7}{2} = 2 \quad \text{and} \quad r = \frac{-3 - 7}{2} = -5 \][/tex]
Thus, the [tex]\( r \)[/tex]-intercepts are:
[tex]\[ (-5, 0), (0, 0), (2, 0) \][/tex]
### Summary:
- The [tex]\( P \)[/tex]-intercept is:
[tex]\[ (0, 0) \][/tex]
- The [tex]\( r \)[/tex]-intercepts are:
[tex]\[ (-5, 0), (0, 0), (2, 0) \][/tex]
Therefore, the solution to the given question is:
Its [tex]\( P \)[/tex]-intercept is [tex]\((0, 0)\)[/tex]
Its [tex]\( r \)[/tex]-intercept(s) is/are [tex]\((-5, 0), (0, 0), (2, 0)\)[/tex]
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