Question

if alpha and beta are zeroes of quadratic polynomial ax²+bx+c, evaluate α³+β³​

Answer 1

Answer:

α³ + ß³ = 3bc/a² - b³/a³

Note:

★ The possible values of variable for which the polynomial becomes zero are called its zeros.

★ A quadratic polynomial can have atmost two zeros.

★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;

Sum of zeros , (α + ß) = -b/a

Product of zeros , αß = c/a

Solution:

Here,

The given quadratic polynomial is :

ax² + bx + c .

Also,

If α and ß are the zeros of the given quadratic polynomial , then ;

α + ß = -b/a

αß = c/a

Now ,

α³ + ß³ = (α + ß)(α² - αß + ß²)

= (α + ß)(α² + 2αß + ß² - 3αß)

= (α + ß)[ (α + ß)² - 3αß ]

= (-b/a)[ (-b/a)² - 3(c/a) ]

= (-b/a)(b²/a² - 3c/a)

= -b³/a³ + 3bc/a²

= 3bc/a² - b³/a³

Hence,

α³ + ß³ = 3bc/a² - b³/a³

Answer 2

Answer:

Step-by-step explanation:

ax²+bx+c

α+β= -b/a= -b/a

αβ=c/a

(α+β)³=α³+β³+3αβ(α+β)

(-b/a)³=α³+β³+3*c/a(-b/a)

-b/a*-b/a*-b/a=α³+β³+3*c/a*-b/a

(-b/a)²=α³+β³+3*c/a

α³+β³=3c/a+(b/a)²

        = 3c+b²/a²