Question Find the value of $a$ if $2-\sqrt{5}$ is a root of a polynomial $w(x)$ , and $2+a$ is also a root.

Studdy Solution

STEP 1

Assumptions
1. $2-\sqrt{5}$ is a root of the polynomial $w(x)$ .
2. $2+a$ is also a root of the polynomial $w(x)$ .
3. The polynomial $w(x)$ has real coefficients.

STEP 2

Since $w(x)$ has real coefficients, the non-real roots must occur in conjugate pairs. This means that if $2-\sqrt{5}$ is a root, then its conjugate $2+\sqrt{5}$ must also be a root.

STEP 3

We compare the given root $2-\sqrt{5}$ with the form of the other root $2+a$ . Since we know that the roots must be conjugates, we can deduce that $a$ must be the positive conjugate of $-\sqrt{5}$ .

STEP 4

Therefore, we have $a = +\sqrt{5}$ .
$a = \sqrt{5}$ is the value that satisfies the condition that $2+a$ is also a root of $w(x)$ .

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