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Answer :
To find the zeros of the function [tex]\( f(x) = -5x^5 - 35x^4 + 70x^3 \)[/tex], we need to solve the equation [tex]\( f(x) = 0 \)[/tex]. Here’s how you can do it step by step:
### Step 1: Factor out the Greatest Common Factor
First, look for the greatest common factor (GCF) in all terms of the polynomial. In [tex]\( -5x^5 - 35x^4 + 70x^3 \)[/tex], each term can be factored by [tex]\( -5x^3 \)[/tex]:
[tex]\[ f(x) = -5x^3(x^2 + 7x - 14) \][/tex]
### Step 2: Solve for the First Zero
The first part, [tex]\(-5x^3\)[/tex], can be set to zero:
[tex]\[ -5x^3 = 0 \][/tex]
This equation simplifies to:
[tex]\[ x^3 = 0 \][/tex]
Therefore:
[tex]\[ x = 0 \][/tex]
Zero is one solution, and it has a multiplicity of 3 due to the [tex]\(x^3\)[/tex].
### Step 3: Solve the Quadratic Part
Next, solve the quadratic equation [tex]\( x^2 + 7x - 14 = 0 \)[/tex]. To find the zeros, use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = -14 \)[/tex]. Plug these values into the formula:
[tex]\[ x = \frac{{-7 \pm \sqrt{{7^2 - 4 \cdot 1 \cdot (-14)}}}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{{-7 \pm \sqrt{{49 + 56}}}}{2} \][/tex]
[tex]\[ x = \frac{{-7 \pm \sqrt{105}}}{2} \][/tex]
This gives us two additional zeros:
[tex]\[ x = \frac{{-7 + \sqrt{105}}}{2} \][/tex]
and
[tex]\[ x = \frac{{-7 - \sqrt{105}}}{2} \][/tex]
### Final Answer
The zeros of the function are:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x = \frac{{-7 + \sqrt{105}}}{2} \)[/tex]
3. [tex]\( x = \frac{{-7 - \sqrt{105}}}{2} \)[/tex]
These zeros indicate where the graph of the polynomial crosses the x-axis.
### Step 1: Factor out the Greatest Common Factor
First, look for the greatest common factor (GCF) in all terms of the polynomial. In [tex]\( -5x^5 - 35x^4 + 70x^3 \)[/tex], each term can be factored by [tex]\( -5x^3 \)[/tex]:
[tex]\[ f(x) = -5x^3(x^2 + 7x - 14) \][/tex]
### Step 2: Solve for the First Zero
The first part, [tex]\(-5x^3\)[/tex], can be set to zero:
[tex]\[ -5x^3 = 0 \][/tex]
This equation simplifies to:
[tex]\[ x^3 = 0 \][/tex]
Therefore:
[tex]\[ x = 0 \][/tex]
Zero is one solution, and it has a multiplicity of 3 due to the [tex]\(x^3\)[/tex].
### Step 3: Solve the Quadratic Part
Next, solve the quadratic equation [tex]\( x^2 + 7x - 14 = 0 \)[/tex]. To find the zeros, use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = -14 \)[/tex]. Plug these values into the formula:
[tex]\[ x = \frac{{-7 \pm \sqrt{{7^2 - 4 \cdot 1 \cdot (-14)}}}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{{-7 \pm \sqrt{{49 + 56}}}}{2} \][/tex]
[tex]\[ x = \frac{{-7 \pm \sqrt{105}}}{2} \][/tex]
This gives us two additional zeros:
[tex]\[ x = \frac{{-7 + \sqrt{105}}}{2} \][/tex]
and
[tex]\[ x = \frac{{-7 - \sqrt{105}}}{2} \][/tex]
### Final Answer
The zeros of the function are:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x = \frac{{-7 + \sqrt{105}}}{2} \)[/tex]
3. [tex]\( x = \frac{{-7 - \sqrt{105}}}{2} \)[/tex]
These zeros indicate where the graph of the polynomial crosses the x-axis.
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