Answer :
To solve the equation [tex]\(28x^3 + 88x^2 - 37x - 7 = 0\)[/tex], knowing that [tex]\(-\frac{7}{2}\)[/tex] is a zero, you can follow these steps:
1. Verification of Known Root:
First, verify that [tex]\(-\frac{7}{2}\)[/tex] is indeed a root of the polynomial. You can substitute it into the equation:
[tex]\[
28\left(-\frac{7}{2}\right)^3 + 88\left(-\frac{7}{2}\right)^2 - 37\left(-\frac{7}{2}\right) - 7 = 0
\][/tex]
After substituting and calculating, the result will be zero, confirming that [tex]\(-\frac{7}{2}\)[/tex] is a root.
2. Polynomial Division:
Since [tex]\(-\frac{7}{2}\)[/tex] is a root, [tex]\((x + \frac{7}{2})\)[/tex] is a factor of the polynomial. We can perform polynomial division to divide the cubic polynomial [tex]\(28x^3 + 88x^2 - 37x - 7\)[/tex] by the factor [tex]\((x + \frac{7}{2})\)[/tex]. After dividing, we obtain a quadratic polynomial.
3. Resulting Quadratic Polynomial:
The division provides the quadratic factor:
[tex]\[
28x^2 - 10x - 2
\][/tex]
4. Solving the Quadratic:
You now solve the quadratic equation [tex]\(28x^2 - 10x - 2 = 0\)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Where [tex]\(a = 28\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -2\)[/tex].
5. Calculating the Roots:
Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-10)^2 - 4 \times 28 \times (-2)
\][/tex]
Solve the equation for the roots, which gives the values:
[tex]\[
x = -0.142857 \quad \text{and} \quad x = 0.5
\][/tex]
6. Final Solution:
Therefore, the solutions (roots) for the equation are:
[tex]\[
x = -\frac{7}{2}, \quad x \approx -0.142857, \quad x = 0.5
\][/tex]
These are the three roots of the given cubic polynomial.
1. Verification of Known Root:
First, verify that [tex]\(-\frac{7}{2}\)[/tex] is indeed a root of the polynomial. You can substitute it into the equation:
[tex]\[
28\left(-\frac{7}{2}\right)^3 + 88\left(-\frac{7}{2}\right)^2 - 37\left(-\frac{7}{2}\right) - 7 = 0
\][/tex]
After substituting and calculating, the result will be zero, confirming that [tex]\(-\frac{7}{2}\)[/tex] is a root.
2. Polynomial Division:
Since [tex]\(-\frac{7}{2}\)[/tex] is a root, [tex]\((x + \frac{7}{2})\)[/tex] is a factor of the polynomial. We can perform polynomial division to divide the cubic polynomial [tex]\(28x^3 + 88x^2 - 37x - 7\)[/tex] by the factor [tex]\((x + \frac{7}{2})\)[/tex]. After dividing, we obtain a quadratic polynomial.
3. Resulting Quadratic Polynomial:
The division provides the quadratic factor:
[tex]\[
28x^2 - 10x - 2
\][/tex]
4. Solving the Quadratic:
You now solve the quadratic equation [tex]\(28x^2 - 10x - 2 = 0\)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Where [tex]\(a = 28\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -2\)[/tex].
5. Calculating the Roots:
Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-10)^2 - 4 \times 28 \times (-2)
\][/tex]
Solve the equation for the roots, which gives the values:
[tex]\[
x = -0.142857 \quad \text{and} \quad x = 0.5
\][/tex]
6. Final Solution:
Therefore, the solutions (roots) for the equation are:
[tex]\[
x = -\frac{7}{2}, \quad x \approx -0.142857, \quad x = 0.5
\][/tex]
These are the three roots of the given cubic polynomial.
