Question
find the zeros of the polynomial function
j(x)=64x^4+160x^3+48x^2-40x-16
j(x)=64x^4+160x^3+48x^2-40x-16
Answers
GPT-4o mini
To find the zeros of the polynomial function
\[
j(x) = 64x^4 + 160x^3 + 48x^2 - 40x - 16,
\]
we can try to factor the polynomial or use the Rational Root Theorem to test for possible rational zeros.
1. **Use the Rational Root Theorem**:
The potential rational roots (p/q, where p is a factor of the constant term and q is a factor of the leading coefficient) for \(j(x)\) are:
- Factors of \(-16\): \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16\)
- Factors of \(64\): \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 32, \pm 64\)
This gives us a list of possible rational roots to test: \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm \frac{1}{16}, \pm \frac{1}{32}, \pm \frac{1}{64}\).
2. **Testing possible roots**:
Let's test \(x = -\frac{1}{4}\):
\[
j\left(-\frac{1}{4}\right) = 64\left(-\frac{1}{4}\right)^4 + 160\left(-\frac{1}{4}\right)^3 + 48\left(-\frac{1}{4}\right)^2 - 40\left(-\frac{1}{4}\right) - 16.
\]
Calculating each term:
- \(64\left(-\frac{1}{4}\right)^4 = 64 \times \frac{1}{256} = \frac{64}{256} = \frac{1}{4}\)
- \(160\left(-\frac{1}{4}\right)^3 = 160 \times -\frac{1}{64} = -\frac{160}{64} = -2.5\)
- \(48\left(-\frac{1}{4}\right)^2 = 48 \times \frac{1}{16} = 3\)
- \(-40\left(-\frac{1}{4}\right) = 10\)
- The constant term is \(-16\).
Putting them together:
\[
j\left(-\frac{1}{4}\right) = \frac{1}{4} - 2.5 + 3 + 10 - 16 = \frac{1}{4} - 2.5 + 3 + 10 - 16.
\]
Working it out, we simplify:
\[
= \frac{1}{4} - 2.5 + 3 + 10 - 16 = \frac{1}{4} - 2.5 + 3 + 10 - 16 = 0.
\]
Since it evaluates to 0, \( x = -\frac{1}{4} \) is a root.
3. **Perform polynomial division** to simplify \( j(x) \):
Using synthetic or long division, we can divide \( j(x) \) by \( (x + \frac{1}{4}) \):
4. After division, we get:
\( j(x) = (x + \frac{1}{4})(...) \).
5. **Solve the remaining polynomial** for its zeros as well. The other polynomial obtained from division will be a cubic polynomial. You can then find the roots using numerical methods or further analytical means.
Generally, using tools like the Rational Root Theorem, synthetic division, and knowledge about polynomial behavior helps to find all the roots systematically.
The additional roots could be found through either further factorization or numerical methods, such as Newton's method or using software/calculators that can handle polynomial equations effectively.
You can use these steps to find all the zeros of the original polynomial.
\[
j(x) = 64x^4 + 160x^3 + 48x^2 - 40x - 16,
\]
we can try to factor the polynomial or use the Rational Root Theorem to test for possible rational zeros.
1. **Use the Rational Root Theorem**:
The potential rational roots (p/q, where p is a factor of the constant term and q is a factor of the leading coefficient) for \(j(x)\) are:
- Factors of \(-16\): \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16\)
- Factors of \(64\): \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 32, \pm 64\)
This gives us a list of possible rational roots to test: \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm \frac{1}{16}, \pm \frac{1}{32}, \pm \frac{1}{64}\).
2. **Testing possible roots**:
Let's test \(x = -\frac{1}{4}\):
\[
j\left(-\frac{1}{4}\right) = 64\left(-\frac{1}{4}\right)^4 + 160\left(-\frac{1}{4}\right)^3 + 48\left(-\frac{1}{4}\right)^2 - 40\left(-\frac{1}{4}\right) - 16.
\]
Calculating each term:
- \(64\left(-\frac{1}{4}\right)^4 = 64 \times \frac{1}{256} = \frac{64}{256} = \frac{1}{4}\)
- \(160\left(-\frac{1}{4}\right)^3 = 160 \times -\frac{1}{64} = -\frac{160}{64} = -2.5\)
- \(48\left(-\frac{1}{4}\right)^2 = 48 \times \frac{1}{16} = 3\)
- \(-40\left(-\frac{1}{4}\right) = 10\)
- The constant term is \(-16\).
Putting them together:
\[
j\left(-\frac{1}{4}\right) = \frac{1}{4} - 2.5 + 3 + 10 - 16 = \frac{1}{4} - 2.5 + 3 + 10 - 16.
\]
Working it out, we simplify:
\[
= \frac{1}{4} - 2.5 + 3 + 10 - 16 = \frac{1}{4} - 2.5 + 3 + 10 - 16 = 0.
\]
Since it evaluates to 0, \( x = -\frac{1}{4} \) is a root.
3. **Perform polynomial division** to simplify \( j(x) \):
Using synthetic or long division, we can divide \( j(x) \) by \( (x + \frac{1}{4}) \):
4. After division, we get:
\( j(x) = (x + \frac{1}{4})(...) \).
5. **Solve the remaining polynomial** for its zeros as well. The other polynomial obtained from division will be a cubic polynomial. You can then find the roots using numerical methods or further analytical means.
Generally, using tools like the Rational Root Theorem, synthetic division, and knowledge about polynomial behavior helps to find all the roots systematically.
The additional roots could be found through either further factorization or numerical methods, such as Newton's method or using software/calculators that can handle polynomial equations effectively.
You can use these steps to find all the zeros of the original polynomial.