Question

If α and β are the zeros of the quadratic polynomial p(x)= x² − 4x + 3, find the value of (α)4(β)³ + (α)³(β)4.

Answer 1

EXPLANATION.

α,β are the zeroes of the quadratic polynomial.

⇒ p(x) = x² - 4x + 3.

As we know that,

Sum of the zeroes of the quadratic equation.

⇒ α + β = -b/a.

⇒ α + β = -(-4)/1 = 4.

Products of the zeroes of the quadratic equation.

⇒ αβ = c/a.

⇒ αβ = 3.

To find:

⇒ (α)⁴(β)³ + α³β⁴.

⇒ α³β³[α + β].

⇒ (αβ)³[α + β].

⇒ (3)³[4].

⇒ 27[4].

⇒ 108.

α⁴β³ + α³β⁴ = 108.

                                                                                                                                             

MORE INFORMATION.

Conjugate Roots.

If D < 0.

One root = α + iβ.

Other root = α - iβ.

If D > 0

One root = α + √β.

Other root = α - √β.

Answer 2

Answer:

We know that

α + β = -b/a.

Now

α + β = -(-4)/1

α + β = 4/1 = 4

Now,

Product of zeroes = α β = c/a

αβ = 3/1 = 3

Now

(α)⁴(β)³ + α³β⁴.

Taking [α + β]. as common

(αβ)³[α + β].

3³ [4].

27 [4]

108