if alpha and bita are the roots if equation x^2+px+q=0 find alphacube+bitacube
Answers
Answer :
α³ + β³ = 3pq - p³
Step-by-step explanation :
➤ Quadratic Polynomials :
✯ It is a polynomial of degree 2
✯ General form :
ax² + bx + c = 0
✯ Determinant, D = b² - 4ac
✯ Based on the value of Determinant, we can define the nature of roots.
D > 0 ; real and unequal roots
D = 0 ; real and equal roots
D < 0 ; no real roots i.e., imaginary
✯ Relationship between zeroes and coefficients :
✩ Sum of zeroes = -b/a
✩ Product of zeroes = c/a
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Given quadratic equation,
x² + px + q = 0
It is of the form ax² + bx + c = 0
a = 1 , b = p , c = q
α and β are the roots of the given equation.
⇒ Sum of roots = -b/a
α + β = -p/1
α + β = -p
⇒ Product of roots = c/a
αβ = q/1
αβ = q
we know,
(x + y)³ = x³ + y³ + 3xy(x + y)
So, (α + β)³ = α³ + β³ + 3αβ(α + β)
α³ + β³ = (α + β)³ - 3αβ(α + β)
= (-p)³ - 3(q)(-p)
= -p³ + 3pq
= 3pq - p³
∴ α³ + β³ = 3pq - p³
Refer the attachment......
