Answer :
To find the zeros of the polynomial [tex]\(2x^4 + 8x^3 - 20x^2 - 20x + 48\)[/tex], we can follow these steps:
1. Identify the Polynomial Form:
The polynomial given is a quartic polynomial (degree 4).
2. Attempt Factoring by Grouping:
First, we might try factoring by grouping, but in this case, let's proceed with a different strategy for a fourth-degree polynomial.
3. Search for Rational Roots:
We use the Rational Root Theorem, which suggests that any rational root of the polynomial [tex]\(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\)[/tex] must be of the form [tex]\(\pm \frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] is a factor of the constant term (48) and [tex]\(q\)[/tex] is a factor of the leading coefficient (2).
The possible rational roots could be [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 48, \pm \frac{1}{2}, \pm \frac{3}{2}\)[/tex].
4. Check These Roots Manually or Use Synthetic Division:
Evaluating each possible root or using synthetic division helps to identify real roots. But after much work, if no rational roots are obvious, more advanced methods are necessary.
5. Use Numerical or Graphical Methods:
For complicated quartic equations, finding exact algebraic solutions analytically can be tricky and time-consuming. Numerical methods or graphing might help to estimate roots.
6. Conclude with Solutions:
The roots are real and are expressed as:
[tex]\[
x \approx -1 + \frac{\sqrt{254/(9a) + 2a + 32/3}}{2} - \frac{\sqrt{-2a - 36/\sqrt{254/(9a) + 2a + 32/3} - 254/(9a) + 64/3}}{2},
\][/tex]
[tex]\[
x \approx -1 + \frac{\sqrt{254/(9a) + 2a + 32/3}}{2} + \frac{\sqrt{-2a - 36/\sqrt{254/(9a) + 2a + 32/3} - 254/(9a) + 64/3}}{2},
\][/tex]
[tex]\[
x \approx -\frac{\sqrt{254/(9a) + 2a + 32/3}}{2} - 1 + \frac{\sqrt{-2a - 254/(9a) + 36/\sqrt{254/(9a) + 2a + 32/3} + 64/3}}{2},
\][/tex]
[tex]\[
x \approx -\frac{\sqrt{254/(9a) + 2a + 32/3}}{2} - \frac{\sqrt{-2a - 254/(9a) + 36/\sqrt{254/(9a) + 2a + 32/3} + 64/3}}{2} - 1
\][/tex]
Although these roots look complex and are given in terms of expressions involving square roots, they are indeed valid solutions. Using numerical methods or algebraic software would give more understandable decimal approximations.
Remember, finding zeros of polynomials, especially of degree 3 or higher, might require advanced algebraic techniques or computational tools for exact results.
1. Identify the Polynomial Form:
The polynomial given is a quartic polynomial (degree 4).
2. Attempt Factoring by Grouping:
First, we might try factoring by grouping, but in this case, let's proceed with a different strategy for a fourth-degree polynomial.
3. Search for Rational Roots:
We use the Rational Root Theorem, which suggests that any rational root of the polynomial [tex]\(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\)[/tex] must be of the form [tex]\(\pm \frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] is a factor of the constant term (48) and [tex]\(q\)[/tex] is a factor of the leading coefficient (2).
The possible rational roots could be [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 48, \pm \frac{1}{2}, \pm \frac{3}{2}\)[/tex].
4. Check These Roots Manually or Use Synthetic Division:
Evaluating each possible root or using synthetic division helps to identify real roots. But after much work, if no rational roots are obvious, more advanced methods are necessary.
5. Use Numerical or Graphical Methods:
For complicated quartic equations, finding exact algebraic solutions analytically can be tricky and time-consuming. Numerical methods or graphing might help to estimate roots.
6. Conclude with Solutions:
The roots are real and are expressed as:
[tex]\[
x \approx -1 + \frac{\sqrt{254/(9a) + 2a + 32/3}}{2} - \frac{\sqrt{-2a - 36/\sqrt{254/(9a) + 2a + 32/3} - 254/(9a) + 64/3}}{2},
\][/tex]
[tex]\[
x \approx -1 + \frac{\sqrt{254/(9a) + 2a + 32/3}}{2} + \frac{\sqrt{-2a - 36/\sqrt{254/(9a) + 2a + 32/3} - 254/(9a) + 64/3}}{2},
\][/tex]
[tex]\[
x \approx -\frac{\sqrt{254/(9a) + 2a + 32/3}}{2} - 1 + \frac{\sqrt{-2a - 254/(9a) + 36/\sqrt{254/(9a) + 2a + 32/3} + 64/3}}{2},
\][/tex]
[tex]\[
x \approx -\frac{\sqrt{254/(9a) + 2a + 32/3}}{2} - \frac{\sqrt{-2a - 254/(9a) + 36/\sqrt{254/(9a) + 2a + 32/3} + 64/3}}{2} - 1
\][/tex]
Although these roots look complex and are given in terms of expressions involving square roots, they are indeed valid solutions. Using numerical methods or algebraic software would give more understandable decimal approximations.
Remember, finding zeros of polynomials, especially of degree 3 or higher, might require advanced algebraic techniques or computational tools for exact results.
