Obtain All The Zeros Of The Polynomial F[x]=x^4-3x^2-x^2+9x-6 If Two Of Its Zeros Are -3 And 3
In the study of polynomial functions, finding all the zeros (roots) of a polynomial is a fundamental task that provides insight into its behavior, graph, and factorization. Given the polynomial \( F[x] = x^4 - 3x^2 - x^2 + 9x - 6 \) and the information that two of its zeros are \(-3\) and \(3\), our goal is to determine all zeros of the polynomial explicitly. This process involves simplifying the polynomial, leveraging the given roots to factor it, and applying algebraic techniques such as synthetic division and quadratic factorization. In this article, we will systematically analyze the polynomial, verify the known zeros, find the remaining zeros, and interpret the results.
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Understanding the Polynomial and Initial Observations
Step 1: Simplify the Polynomial Expression
The original polynomial is:
\[
F[x] = x^4 - 3x^2 - x^2 + 9x - 6
\]
Combine like terms:
\[
F[x] = x^4 - (3x^2 + x^2) + 9x - 6 = x^4 - 4x^2 + 9x - 6
\]
Thus, the simplified form is:
\[
F[x] = x^4 - 4x^2 + 9x - 6
\]
Observation: The polynomial is a quartic with mixed degrees, making direct factorization challenging. However, the given zeros \(-3\) and \(3\) suggest that the factors \((x + 3)\) and \((x - 3)\) are roots, which hints at possible quadratic factors associated with these roots.
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Using Known Zeros to Factor the Polynomial
Step 2: Construct the Factors from Known Zeros
Since \(-3\) and \(3\) are roots, the corresponding factors are:
\[
(x + 3) \quad \text{and} \quad (x - 3)
\]
Their product gives:
\[
(x + 3)(x - 3) = x^2 - 9
\]
This suggests that \(x^2 - 9\) divides the polynomial \(F[x]\).
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Step 3: Polynomial Division to Find the Remaining Factor
To verify this, perform polynomial division of \(F[x]\) by \(x^2 - 9\). The goal is to find a quadratic quotient \(Q(x)\) such that:
\[
F[x] = (x^2 - 9) \times Q(x)
\]
Performing Polynomial Division:
Divide \(x^4 - 4x^2 + 9x - 6\) by \(x^2 - 9\).
Set up the division:
- Leading term division: \(x^4 \div x^2 = x^2\)
- Multiply back: \(x^2 \times (x^2 - 9) = x^4 - 9x^2\)
- Subtract: \((x^4 - 4x^2 + 9x - 6) - (x^4 - 9x^2) = 5x^2 + 9x - 6\)
Next:
- Divide \(5x^2\) by \(x^2\): \(5\)
- Multiply: \(5 \times (x^2 - 9) = 5x^2 - 45\)
- Subtract: \((5x^2 + 9x - 6) - (5x^2 - 45) = 9x + 39\)
Since the degree of the remainder \(9x + 39\) is less than 2, the division process stops.
The quotient is:
\[
Q(x) = x^2 + 5
\]
and the remainder is:
\[
R(x) = 9x + 39
\]
Because the remainder is not zero, \(x^2 - 9\) is not a factor of \(F[x]\).
Implication: Our initial assumption that \(x^2 - 9\) divides \(F[x]\) exactly is invalid, indicating that the roots \(-3\) and \(3\) are roots of the polynomial, but the polynomial is not divisible by \(x^2 - 9\). Instead, these roots are roots of the polynomial, but the polynomial's factorization is not directly through quadratic factors of the form \(x^2 - 9\).
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Refined Approach: Polynomial Synthetic Division and Factoring
Step 4: Synthetic Division with Known Roots
Given roots \(-3\) and \(3\), we can perform synthetic division to factor out \((x + 3)\) and \((x - 3)\) sequentially.
Performing synthetic division by \(x = 3\):
Coefficients of \(F[x]\):
\[
1 \quad 0 \quad -4 \quad 9 \quad -6
\]
- Bring down the 1.
- Multiply by 3: \(1 \times 3 = 3\); add: \(0 + 3 = 3\).
- Multiply by 3: \(3 \times 3 = 9\); add: \(-4 + 9 = 5\).
- Multiply by 3: \(5 \times 3 = 15\); add: \(9 + 15 = 24\).
- Multiply by 3: \(24 \times 3 = 72\); add: \(-6 + 72 = 66\).
The final remainder is 66, not zero, indicating that \(x=3\) is not a root unless the division yields zero remainder.
Similarly, synthetic division by \(x = -3\):
Coefficients:
\[
1 \quad 0 \quad -4 \quad 9 \quad -6
\]
- Bring down 1.
- Multiply by \(-3\): \(1 \times -3 = -3\); add: \(0 + (-3) = -3\).
- Multiply by \(-3\): \(-3 \times -3 = 9\); add: \(-4 + 9 = 5\).
- Multiply by \(-3\): \(5 \times -3 = -15\); add: \(9 + (-15) = -6\).
- Multiply by \(-3\): \(-6 \times -3 = 18\); add: \(-6 + 18 = 12\).
Remainder is 12, again not zero.
Conclusion: The division indicates that \(-3\) and \(3\) are not roots of this polynomial, contradicting the initial assumption. This suggests a need to verify the initial problem statement or the polynomial.
---
Re-examination of the Polynomial and Roots
Step 5: Verify the Polynomial and Roots
The given polynomial is:
\[
F[x] = x^4 - 3x^2 - x^2 + 9x - 6
\]
which simplifies to:
\[
F[x] = x^4 - 4x^2 + 9x - 6
\]
Suppose the roots are \(-3\) and \(3\). Let's test these roots directly:
- \(F[-3]\):
\[
(-3)^4 - 4(-3)^2 + 9(-3) - 6 = 81 - 4(9) - 27 -6 = 81 - 36 - 27 - 6 = 81 - 36 - 33 = 81 - 69 = 12
\]
Not zero, so \(-3\) is not a root.
- \(F[3]\):
\[
81 - 36 + 27 - 6 = 81 - 36 + 21 = 81 - 15 = 66
\]
Not zero either.
Implication: The initial statement that the zeros are \(-3\) and \(3\) appears inconsistent with the polynomial as written, unless the polynomial is different.
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Clarification and Corrected Approach
Given the inconsistency, the most logical conclusion is that the polynomial was intended to be:
\[
F[x] = x^4 - 3x^2 + 9x - 6
\]
which simplifies to:
\[
F[x] = x^4 - 3x^2 + 9x - 6
\]
and the roots \(-3\) and \(3\).
Let's verify if \(-3\) and \(3\) are roots of this polynomial:
- \(F[-3]\):
\[
(-3)^4 - 3(-3)^2 + 9(-3) - 6 = 81 - 3(9) - 27 - 6 = 81 - 27 - 27 - 6 = 81 - 60 =
