Question

1If α and β are the zeroes of a polynomial such that α + β = -6 and αβ = 5, then find the polynomial.

Answer 1

Given: It is given that, α and β are the zeroes of a polynomial such that;

• α + β = – 6
• αβ = 5

Need to find: The Quadratic Polynomial$?$

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As we know that,

The Quadratic Polynomial whose zeroes are α and β respectively is Given by :

⠀⠀⠀★ f ( x ) = k( x² – (α + β)x + αβ ) ★

where,

• (α + β) = Sum of Zeroes
• (αβ) = Product of Zeroes
• k = Non – zero real no.

• Q u a d r a t i c⠀P o l y n o m i a l :

↠ k(x² – (– 6)x + 5

↠ k(x² + 6x + 5)

↠ x² + 6x + 5

∴ Hence, the Quadratic Polynomial is x² + 6x + 5.

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• M o r e T o K n o w :

• A Quadratic Polynomial is a polynomial which is in the form of (ax² + bx + c). Here, a, b & c are any real numbers.
• a is the Coefficient of x²
• b is the Coefficient of x
• & c is any Constant no.

» Sum of any Quadratic Polynomial is given by : (α + β) = – b/a

» Product of any Quadratic Polynomial is given by : (αβ) = c/a

Answer 2

Given :-

If α and β are the zeroes of a polynomial such that α + β = -6 and αβ = 5,

To Find :-

Polynomial

Solution :-

We know that

Standard form of a quadratic polynomial = x² - (α + β)x + αβ

Sum of zeroes = α + β

Sum of zeroes = -6 (1)

Product of zeroes = αβ

Product of zeroes = 5 (2)

Putting value from 1 and

x² - (-6)x + 5

x² + 6x + 5

Verification

x² + 6x + 5

x² + 5x + x + 5

x(x + 5) + 1(x + 5)

(x + 5)(x + 1)

x = -5 & x = -1

α + β = -b/a

-1 + (-5) = -(6)/1

-1 - 5 = -6

-6 = -6

αβ = c/a

-1 × (-5) = 5/1

5 = 5