Answer :
To solve the polynomial [tex]\(x^3 + 5x^2 - 9x - 45\)[/tex], we will find its roots, its derivative, and the value of the polynomial when [tex]\(x = 0\)[/tex].
### Step 1: Find the roots of the polynomial
The roots of the polynomial are the values of [tex]\(x\)[/tex] for which the polynomial equals zero. In this case, the roots are:
1. [tex]\(x = -5\)[/tex]
2. [tex]\(x = -3\)[/tex]
3. [tex]\(x = 3\)[/tex]
These are the solutions for the equation [tex]\(x^3 + 5x^2 - 9x - 45 = 0\)[/tex].
### Step 2: Find the derivative of the polynomial
The derivative of a polynomial is found by applying the power rule to each term. For a term in the form [tex]\(ax^n\)[/tex], the derivative is [tex]\(nax^{n-1}\)[/tex]. Applying this to each term:
- The derivative of [tex]\(x^3\)[/tex] is [tex]\(3x^2\)[/tex].
- The derivative of [tex]\(5x^2\)[/tex] is [tex]\(10x\)[/tex].
- The derivative of [tex]\(-9x\)[/tex] is [tex]\(-9\)[/tex].
- The derivative of [tex]\(-45\)[/tex], being a constant, is [tex]\(0\)[/tex].
Thus, the derivative of the polynomial is:
[tex]\[3x^2 + 10x - 9\][/tex]
### Step 3: Evaluate the polynomial at [tex]\(x = 0\)[/tex]
To find the value of the polynomial at [tex]\(x = 0\)[/tex], substitute [tex]\(0\)[/tex] for [tex]\(x\)[/tex] in the expression:
[tex]\[x^3 + 5x^2 - 9x - 45 = 0^3 + 5(0)^2 - 9(0) - 45\][/tex]
Simplifying, we find that:
[tex]\[= 0 + 0 - 0 - 45 = -45\][/tex]
Therefore, the value of the polynomial when [tex]\(x = 0\)[/tex] is [tex]\(-45\)[/tex].
Summary:
- The roots of the polynomial are [tex]\(-5\)[/tex], [tex]\(-3\)[/tex], and [tex]\(3\)[/tex].
- The derivative is [tex]\(3x^2 + 10x - 9\)[/tex].
- The value at [tex]\(x = 0\)[/tex] is [tex]\(-45\)[/tex].
### Step 1: Find the roots of the polynomial
The roots of the polynomial are the values of [tex]\(x\)[/tex] for which the polynomial equals zero. In this case, the roots are:
1. [tex]\(x = -5\)[/tex]
2. [tex]\(x = -3\)[/tex]
3. [tex]\(x = 3\)[/tex]
These are the solutions for the equation [tex]\(x^3 + 5x^2 - 9x - 45 = 0\)[/tex].
### Step 2: Find the derivative of the polynomial
The derivative of a polynomial is found by applying the power rule to each term. For a term in the form [tex]\(ax^n\)[/tex], the derivative is [tex]\(nax^{n-1}\)[/tex]. Applying this to each term:
- The derivative of [tex]\(x^3\)[/tex] is [tex]\(3x^2\)[/tex].
- The derivative of [tex]\(5x^2\)[/tex] is [tex]\(10x\)[/tex].
- The derivative of [tex]\(-9x\)[/tex] is [tex]\(-9\)[/tex].
- The derivative of [tex]\(-45\)[/tex], being a constant, is [tex]\(0\)[/tex].
Thus, the derivative of the polynomial is:
[tex]\[3x^2 + 10x - 9\][/tex]
### Step 3: Evaluate the polynomial at [tex]\(x = 0\)[/tex]
To find the value of the polynomial at [tex]\(x = 0\)[/tex], substitute [tex]\(0\)[/tex] for [tex]\(x\)[/tex] in the expression:
[tex]\[x^3 + 5x^2 - 9x - 45 = 0^3 + 5(0)^2 - 9(0) - 45\][/tex]
Simplifying, we find that:
[tex]\[= 0 + 0 - 0 - 45 = -45\][/tex]
Therefore, the value of the polynomial when [tex]\(x = 0\)[/tex] is [tex]\(-45\)[/tex].
Summary:
- The roots of the polynomial are [tex]\(-5\)[/tex], [tex]\(-3\)[/tex], and [tex]\(3\)[/tex].
- The derivative is [tex]\(3x^2 + 10x - 9\)[/tex].
- The value at [tex]\(x = 0\)[/tex] is [tex]\(-45\)[/tex].
