Question

If α and β are zeros of the polynomial p(x) =3x2

-10x+7 then find the value of α3+β3​

Answer 1

EXPLANATION.

α and β are the zeroes of the polynomial.

⇒ 3x² - 10x + 7 = 0.

As we know that,

Sum of the zeroes of the quadratic equation.

⇒ α + β = -b/a.

⇒ α + β = -(-10)/3 = 10/3.

Products of the zeroes of the quadratic equation.

⇒ αβ = c/a.

⇒ αβ = 7/3.

To find = (α³ + β³).

As we know that,

Formula of :

⇒ x³ + y³ = (x + y)(x² - xy + y²).

⇒ x² + y² = (x + y)² - 2xy.

⇒ (α³ + β³) = (α + β)(α² + β² - αβ).

⇒ (α³ + β³) = (α + β)[(α + β)² - 2αβ - αβ].

⇒ (α³ + β³) = (α + β)[(α + β)² - 3αβ].

Put the values in the equation, we get.

⇒ (α³ + β³) = (10/3)[(10/3)² - 3(7/3)].

⇒ (α³ + β³) = (10/3)[100/9 - 7].

⇒ (α³ + β³) = (10/3)[100 - 63/9].

⇒ (α³ + β³) = (10/3)[37/9].

⇒ (α³ + β³) = 370/27.

                                                                                                                         

MORE INFORMATION.

Nature of the factors of the quadratic expression.

(1) = Real and different, if b² - 4ac > 0.

(2) = Rational and different, if b² - 4ac is a perfect square.

(3) = Real and equal, if b² - 4ac = 0.

(4) = If D < 0 Roots are imaginary and unequal or complex conjugate.

Answer 2

$\huge\bf{Answer :}$

Given :-

• Given quadratic equation, p(x) = 3x²-10x+7
• Zeroes of the p(x) = α and β

To find :-

• The value of α³+β³

Explanation :-

On comparing p(x) with ax²+bx+c, we get :-

• a = 3
• b = -10, and
• c = 7

Now, sum of the roots of p(x) = -(coefficient of x)/coefficient of x²

⇒ α + β = - b / a

∴ α + β = - ( - 10 ) / 3 ⇒ 10/3 --------(i)

And, product of the roots of p(x) = constant term / coefficient of x²

⇒ α.β = 7 / 3 -------(ii)

Here, we're asked to find the sum of cubes of roots of p(x), i.e. α³+β³.

We know that,

• a³+b³ = (a+b)(a²+b²-ab), and
• a²+b² = (a+b)² - 2a.b

So, in terms of α and β, we have,

• α³+β³ = (α+β)(α²+β²-α.β), and
• α²+β² = (α+β)²-2αβ

Now, (α³+β³) = (10/3){(10/3)²-3×(7/3)} ------ [ from (i) and (ii) ]

⇒ (10/3)(100-63/8) = (10/3×37/9)

⇒ 370/27 ✔️

Hence, the required value of (α³+β³) is 370/27.

$\huge\bf{Extra\: Information :}$

1) Standard form of a cubic polynomial : ax³+bx²+cx+d , where,

• a = coefficient of x³
• b = coefficient of x²
• c = coefficient of x, and
• d = constant term.

2) Roots of a cubic equation = γ, α and β

3) Relation of roots with the coefficients of the cubic polynomial :-

a) Products of the roots =α×β×γ= -d/a

b) Sum of the roots =α+β+γ = -b/a, and

c) Product of roots taken two at a time = c/a

[ Note = refer to 1) for knowing the values of a,b,c and d ]

4) Formula for finding quadratic equation :

⇒ x²-( sum of zeroes)x + (product of zeroes)

⇒ x² - (α+β) + (αβ)

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