Answer :
Certainly! To find the zeros of the polynomial function [tex]\( f(x) = x^3 + 7x^2 - 9x - 63 \)[/tex] and to determine the multiplicity of each zero, we can follow these detailed steps.
### Step 1: Identify Possible Rational Zeros
We can use the Rational Root Theorem to identify possible rational zeros. The Rational Root Theorem states that any potential rational zero [tex]\( p/q \)[/tex] of the polynomial [tex]\( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \)[/tex] must be a factor of the constant term [tex]\( a_0 \)[/tex] divided by a factor of the leading coefficient [tex]\( a_n \)[/tex].
For the polynomial [tex]\( f(x) = x^3 + 7x^2 - 9x - 63 \)[/tex]:
- The constant term [tex]\( a_0 = -63 \)[/tex].
- The leading coefficient [tex]\( a_n = 1 \)[/tex].
The possible rational zeros are the factors of [tex]\(-63\)[/tex]: [tex]\(\pm 1, \pm 3, \pm 7, \pm 9, \pm 21, \pm 63\)[/tex].
### Step 2: Test Potential Zeros
We test these potential zeros by substituting them into the polynomial [tex]\( f(x) \)[/tex] to see if they result in [tex]\( f(x) = 0 \)[/tex].
Let’s start with [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 3^3 + 7(3^2) - 9(3) - 63 = 27 + 63 - 27 - 63 = 0 \][/tex]
Since [tex]\( f(3) = 0 \)[/tex], [tex]\( x = 3 \)[/tex] is a zero of the polynomial.
### Step 3: Perform Polynomial Division
We will now factor [tex]\( (x - 3) \)[/tex] out of the polynomial using polynomial division or synthetic division.
Using synthetic division to divide [tex]\( x^3 + 7x^2 - 9x - 63 \)[/tex] by [tex]\( x - 3 \)[/tex]:
[tex]\[ \begin{array}{r|rrrr} 3 & 1 & 7 & -9 & -63 \\ & & 3 & 30 & 63 \\ \hline & 1 & 10 & 21 & 0 \\ \end{array} \][/tex]
The quotient is [tex]\( x^2 + 10x + 21 \)[/tex]. So, we now have:
[tex]\[ f(x) = (x - 3)(x^2 + 10x + 21) \][/tex]
### Step 4: Factor the Quadratic
Next, we factor the quadratic [tex]\( x^2 + 10x + 21 \)[/tex].
The quadratic [tex]\( x^2 + 10x + 21 \)[/tex] can be factored into:
[tex]\[ x^2 + 10x + 21 = (x + 3)(x + 7) \][/tex]
Therefore, the complete factorization of the polynomial [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = (x - 3)(x + 3)(x + 7) \][/tex]
### Step 5: Determine Zeros and Their Multiplicities
From the factorized form [tex]\( f(x) = (x - 3)(x + 3)(x + 7) \)[/tex], we can identify the zeros and their multiplicities:
- [tex]\( x = 3 \)[/tex] has a multiplicity of 1.
- [tex]\( x = -3 \)[/tex] has a multiplicity of 1.
- [tex]\( x = -7 \)[/tex] has a multiplicity of 1.
### Summary
The zeros of the polynomial [tex]\( f(x) = x^3 + 7x^2 - 9x - 63 \)[/tex] and their multiplicities are:
[tex]\[ \begin{align*} x = 3 & \quad \text{(multiplicity 1)} \\ x = -3 & \quad \text{(multiplicity 1)} \\ x = -7 & \quad \text{(multiplicity 1)} \end{align*} \][/tex]
### Step 1: Identify Possible Rational Zeros
We can use the Rational Root Theorem to identify possible rational zeros. The Rational Root Theorem states that any potential rational zero [tex]\( p/q \)[/tex] of the polynomial [tex]\( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \)[/tex] must be a factor of the constant term [tex]\( a_0 \)[/tex] divided by a factor of the leading coefficient [tex]\( a_n \)[/tex].
For the polynomial [tex]\( f(x) = x^3 + 7x^2 - 9x - 63 \)[/tex]:
- The constant term [tex]\( a_0 = -63 \)[/tex].
- The leading coefficient [tex]\( a_n = 1 \)[/tex].
The possible rational zeros are the factors of [tex]\(-63\)[/tex]: [tex]\(\pm 1, \pm 3, \pm 7, \pm 9, \pm 21, \pm 63\)[/tex].
### Step 2: Test Potential Zeros
We test these potential zeros by substituting them into the polynomial [tex]\( f(x) \)[/tex] to see if they result in [tex]\( f(x) = 0 \)[/tex].
Let’s start with [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 3^3 + 7(3^2) - 9(3) - 63 = 27 + 63 - 27 - 63 = 0 \][/tex]
Since [tex]\( f(3) = 0 \)[/tex], [tex]\( x = 3 \)[/tex] is a zero of the polynomial.
### Step 3: Perform Polynomial Division
We will now factor [tex]\( (x - 3) \)[/tex] out of the polynomial using polynomial division or synthetic division.
Using synthetic division to divide [tex]\( x^3 + 7x^2 - 9x - 63 \)[/tex] by [tex]\( x - 3 \)[/tex]:
[tex]\[ \begin{array}{r|rrrr} 3 & 1 & 7 & -9 & -63 \\ & & 3 & 30 & 63 \\ \hline & 1 & 10 & 21 & 0 \\ \end{array} \][/tex]
The quotient is [tex]\( x^2 + 10x + 21 \)[/tex]. So, we now have:
[tex]\[ f(x) = (x - 3)(x^2 + 10x + 21) \][/tex]
### Step 4: Factor the Quadratic
Next, we factor the quadratic [tex]\( x^2 + 10x + 21 \)[/tex].
The quadratic [tex]\( x^2 + 10x + 21 \)[/tex] can be factored into:
[tex]\[ x^2 + 10x + 21 = (x + 3)(x + 7) \][/tex]
Therefore, the complete factorization of the polynomial [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = (x - 3)(x + 3)(x + 7) \][/tex]
### Step 5: Determine Zeros and Their Multiplicities
From the factorized form [tex]\( f(x) = (x - 3)(x + 3)(x + 7) \)[/tex], we can identify the zeros and their multiplicities:
- [tex]\( x = 3 \)[/tex] has a multiplicity of 1.
- [tex]\( x = -3 \)[/tex] has a multiplicity of 1.
- [tex]\( x = -7 \)[/tex] has a multiplicity of 1.
### Summary
The zeros of the polynomial [tex]\( f(x) = x^3 + 7x^2 - 9x - 63 \)[/tex] and their multiplicities are:
[tex]\[ \begin{align*} x = 3 & \quad \text{(multiplicity 1)} \\ x = -3 & \quad \text{(multiplicity 1)} \\ x = -7 & \quad \text{(multiplicity 1)} \end{align*} \][/tex]
