Certainly! Let's solve the problem step by step.
Step 1: Given conditions for the roots
We are provided with the following conditions for the roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex]:
1. [tex]\(\alpha + \beta = 24\)[/tex]
2. [tex]\(\alpha - \beta = 8\)[/tex]
Step 2: Solve for [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex]
To find the values of [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex], we can use the given conditions:
[tex]\[
\alpha + \beta = 24
\][/tex]
[tex]\[
\alpha - \beta = 8
\][/tex]
We can add these two equations to eliminate [tex]\(\beta\)[/tex]:
[tex]\[
(\alpha + \beta) + (\alpha - \beta) = 24 + 8
\][/tex]
[tex]\[
2\alpha = 32
\][/tex]
[tex]\[
\alpha = 16
\][/tex]
Next, we substitute [tex]\(\alpha = 16\)[/tex] back into the first equation to find [tex]\(\beta\)[/tex]:
[tex]\[
\alpha + \beta = 24
\][/tex]
[tex]\[
16 + \beta = 24
\][/tex]
[tex]\[
\beta = 24 - 16
\][/tex]
[tex]\[
\beta = 8
\][/tex]
Now we have [tex]\(\alpha = 16\)[/tex] and [tex]\(\beta = 8\)[/tex].
Step 3: Form the quadratic polynomial
The standard form of a quadratic polynomial with roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] is:
[tex]\[
P(x) = a(x - \alpha)(x - \beta)
\][/tex]
Since we typically use [tex]\(a = 1\)[/tex] for the simplest polynomial, it simplifies to:
[tex]\[
P(x) = (x - \alpha)(x - \beta)
\][/tex]
Substitute [tex]\(\alpha = 16\)[/tex] and [tex]\(\beta = 8\)[/tex] into the polynomial:
[tex]\[
P(x) = (x - 16)(x - 8)
\][/tex]
Step 4: Expand the polynomial
Let's expand the expression:
[tex]\[
P(x) = (x - 16)(x - 8)
\][/tex]
[tex]\[
P(x) = x^2 - 8x - 16x + 128
\][/tex]
[tex]\[
P(x) = x^2 - 24x + 128
\][/tex]
Therefore, the quadratic polynomial with roots [tex]\(\alpha = 16\)[/tex] and [tex]\(\beta = 8\)[/tex] is:
[tex]\[
P(x) = x^2 - 24x + 128
\][/tex]
This is the quadratic polynomial having [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] as its roots.
