Answer :
To solve the equation [tex]\(36x^3 + 45x^2 - 7x - 4 = 0\)[/tex], given that [tex]\(-\frac{4}{3}\)[/tex] is a zero of the polynomial, we can use polynomial division.
1. Understanding Zero of the Polynomial:
Since [tex]\(-\frac{4}{3}\)[/tex] is a zero, this means that when [tex]\(x = -\frac{4}{3}\)[/tex], the polynomial [tex]\(f(x)\)[/tex] evaluates to zero. Therefore, [tex]\((x + \frac{4}{3})\)[/tex] is a factor of the polynomial.
2. Polynomial Division:
Our goal is to divide the polynomial [tex]\(36x^3 + 45x^2 - 7x - 4\)[/tex] by [tex]\((x + \frac{4}{3})\)[/tex].
3. Perform the Division:
After performing the polynomial division, we find that the quotient is [tex]\(36x^2 - 3x - 3\)[/tex] and the remainder is 0.
4. Understanding the Result:
The division resulted in a quotient of [tex]\(36x^2 - 3x - 3\)[/tex] with no remainder. This confirms that [tex]\((x + \frac{4}{3})\)[/tex] is indeed a factor of the polynomial. We can now rewrite the original polynomial as:
[tex]\[
f(x) = (x + \frac{4}{3})(36x^2 - 3x - 3)
\][/tex]
5. Finding the Remaining Zeros:
To find the other zeros of the polynomial, solve the quadratic equation [tex]\(36x^2 - 3x - 3 = 0\)[/tex].
6. Conclusion:
Solving [tex]\(36x^2 - 3x - 3 = 0\)[/tex] will give the other roots of the polynomial. You can use the quadratic formula where [tex]\(a = 36\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -3\)[/tex] to find these roots. This step completes the solution process for the given polynomial equation.
1. Understanding Zero of the Polynomial:
Since [tex]\(-\frac{4}{3}\)[/tex] is a zero, this means that when [tex]\(x = -\frac{4}{3}\)[/tex], the polynomial [tex]\(f(x)\)[/tex] evaluates to zero. Therefore, [tex]\((x + \frac{4}{3})\)[/tex] is a factor of the polynomial.
2. Polynomial Division:
Our goal is to divide the polynomial [tex]\(36x^3 + 45x^2 - 7x - 4\)[/tex] by [tex]\((x + \frac{4}{3})\)[/tex].
3. Perform the Division:
After performing the polynomial division, we find that the quotient is [tex]\(36x^2 - 3x - 3\)[/tex] and the remainder is 0.
4. Understanding the Result:
The division resulted in a quotient of [tex]\(36x^2 - 3x - 3\)[/tex] with no remainder. This confirms that [tex]\((x + \frac{4}{3})\)[/tex] is indeed a factor of the polynomial. We can now rewrite the original polynomial as:
[tex]\[
f(x) = (x + \frac{4}{3})(36x^2 - 3x - 3)
\][/tex]
5. Finding the Remaining Zeros:
To find the other zeros of the polynomial, solve the quadratic equation [tex]\(36x^2 - 3x - 3 = 0\)[/tex].
6. Conclusion:
Solving [tex]\(36x^2 - 3x - 3 = 0\)[/tex] will give the other roots of the polynomial. You can use the quadratic formula where [tex]\(a = 36\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -3\)[/tex] to find these roots. This step completes the solution process for the given polynomial equation.
