To solve for the value of [tex]\( k \)[/tex] in the polynomial [tex]\( 3x^2 - 12x + k \)[/tex] with zeros [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] such that [tex]\( \alpha - \beta = 2 \)[/tex], we can follow these steps:
1. Express the sum and product of roots using Vieta's formulas:
- Sum of roots [tex]\( \alpha + \beta \)[/tex]:
According to Vieta's formulas, the sum of the roots of the polynomial [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( -b/a \)[/tex]. Here, [tex]\( a = 3 \)[/tex] and [tex]\( b = -12 \)[/tex], so:
[tex]\[
\alpha + \beta = \frac{-(-12)}{3} = \frac{12}{3} = 4
\][/tex]
- Product of roots [tex]\( \alpha \beta \)[/tex]:
According to Vieta's formulas, the product of the roots of the polynomial [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( c/a \)[/tex]. Here, [tex]\( c = k \)[/tex] and [tex]\( a = 3 \)[/tex], so:
[tex]\[
\alpha \beta = \frac{k}{3}
\][/tex]
2. Set up the equations using the given conditions:
- From the given condition, [tex]\( \alpha - \beta = 2 \)[/tex].
- From Vieta's formulas, [tex]\( \alpha + \beta = 4 \)[/tex].
3. Solve for [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:
- Add the equations [tex]\( \alpha - \beta = 2 \)[/tex] and [tex]\( \alpha + \beta = 4 \)[/tex]:
[tex]\[
(\alpha - \beta) + (\alpha + \beta) = 2 + 4
\][/tex]
[tex]\[
2\alpha = 6
\][/tex]
[tex]\[
\alpha = 3
\][/tex]
- Subtract the equation [tex]\( \alpha - \beta = 2 \)[/tex] from [tex]\( \alpha + \beta = 4 \)[/tex]:
[tex]\[
(\alpha + \beta) - (\alpha - \beta) = 4 - 2
\][/tex]
[tex]\[
2\beta = 2
\][/tex]
[tex]\[
\beta = 1
\][/tex]
4. Determine the value of [tex]\( k \)[/tex]:
- Using the product of the roots [tex]\( \alpha \beta = \frac{k}{3} \)[/tex]:
[tex]\[
3 \cdot 1 = \frac{k}{3}
\][/tex]
[tex]\[
k = 3 \cdot (\alpha \beta)
\][/tex]
[tex]\[
k = 3 \cdot (3 \cdot 1) = 9
\][/tex]
Thus, the value of [tex]\( k \)[/tex] is [tex]\( 9 \)[/tex].
