Answer :
Answer:
189
Explanation:
Hello dear friend,
For a quadratic polynomial [tex]\( f(t) = at^2 + bt + c \)[/tex], with zeros [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex], the sum of the zeros is given by [tex]\( \alpha +\beta = -\frac{b}{a} \)[/tex], and the product of the zeros is given by [tex]\( \alpha \beta = \frac{c}{a} \)[/tex].
From Vieta's formulas:
- Sum of the zeros: [tex]\( \alpha + \beta = -(-4) = 4 \)[/tex]
- Product of the zeros: [tex]\( \alpha \beta = 3 \)[/tex]
Now, we need to find [tex]\( \alpha^4 \beta^3 + \alpha^3 \beta^4 \)[/tex].
We can rewrite [tex]\( \alpha^4 \beta^3 + \alpha^3 \beta^4 \) as \( \alpha^3 \beta^3 (\alpha + \beta) + \alpha^3 \beta^3 (\alpha \beta) \)[/tex].
Substituting the known values:
[tex]\( = 3^3 \cdot 4 + 3^3 \cdot 3 \)[/tex]
Calculating:
[tex]\( = 27 \cdot 4 + 27 \cdot 3 \)[/tex]
[tex]\( = 108 + 81 \)[/tex]
[tex]\( = 189 \)[/tex]
So, the value of [tex]\( \alpha^4 \beta^3 + \alpha^3 \beta^4 \)[/tex] is 189.
Do let me know if there are any mistakes as I'm only human. Every human makes mistakes, it's okay :p
I hope you have understood my explanation and felt it was somewhat helpful to you.
Have a great day ahead !