Sure, let's solve this step by step.
Step 1: Identify the zeroes of the given quadratic polynomial [tex]\( f(x) = x^2 - x - 2 \)[/tex]
Given the polynomial [tex]\( f(x) = x^2 - x - 2 \)[/tex], we start by finding its roots (zeroes), [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex].
The roots of the polynomial [tex]\( f(x) = x^2 - x - 2 \)[/tex] are found to be:
- [tex]\(\alpha = -1\)[/tex]
- [tex]\(\beta = 2\)[/tex]
Step 2: Transform the roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] into new roots
We need to find a polynomial whose zeroes are [tex]\(2\alpha + 1\)[/tex] and [tex]\(2\beta + 1\)[/tex].
Using the identified zeroes:
- For [tex]\(\alpha = -1\)[/tex], [tex]\( 2\alpha + 1 = 2(-1) + 1 = -2 + 1 = -1 \)[/tex]
- For [tex]\(\beta = 2\)[/tex], [tex]\( 2\beta + 1 = 2(2) + 1 = 4 + 1 = 5 \)[/tex]
Step 3: Construct the new polynomial from these new zeroes
If [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are the roots of a quadratic polynomial, the polynomial can be expressed as:
[tex]\[ (x - p)(x - q) \][/tex]
For our new roots [tex]\( -1 \)[/tex] and [tex]\( 5 \)[/tex]:
[tex]\[ (x - (-1))(x - 5) = (x + 1)(x - 5) \][/tex]
Step 4: Expand the polynomial
Expanding the polynomial [tex]\( (x + 1)(x - 5) \)[/tex]:
[tex]\[ (x + 1)(x - 5) = x(x - 5) + 1(x - 5) = x^2 - 5x + x - 5 = x^2 - 4x - 5 \][/tex]
Thus, the polynomial whose zeroes are [tex]\(2\alpha + 1\)[/tex] and [tex]\(2\beta + 1\)[/tex] is:
[tex]\[ \boxed{x^2 - 4x - 5} \][/tex]
