Question

find zeros of the polyomial 5x+6x+1 and verify the relation between zeros and coefficient
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Answer 1

Correct Question :

find zeroes of the polynomial 5x²+6x+1 and verify the relation between zeroes and coefficients.

Answer :

-1 and -1/5 are the zeroes.

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

______________________________

Given polynomial,

 5x² + 6x + 1

x² coefficient = 5

x  coefficient = 6

constant term = 1

⇒ Factorize :

  5x² + 6x + 1 = 0

5x² + 5x + x + 1 = 0

5x(x + 1) + 1(x + 1) = 0

 (x + 1) (5x + 1) = 0

=> x + 1 = 0 ; x = -1

=> 5x + 1 = 0 ; x = -1/5

∴ 1 and -1/5 are the zeroes of the polynomial 5x² + 6x + 1

Relation between zeroes and coefficients :

⇒ Sum of zeroes

               $=-1+(\frac{-1}{5}) \\\\ =-1-\frac{1}{5} \\\\ =\frac{-5-1}{5} \\\\ =\frac{-6}{5} \\\\ \bf =\frac{-(x \ coefficient)}{x^2 \ coefficient}$

⇒ Product of zeroes

                $=(-1) \times (\frac{-1}{5}) \\\\ =\frac{1}{5} \\\\ \bf =\frac{constant \ term}{x^2 \ coefficient}$

Hence verified!

Answer 2

refer the attachment.........