Question

find the polynomials given two zeroes 1/4 -1​

Answer 1

Answer :

4x² + 3x - 1

Step-by-step explanation :

➤ Quadratic Polynomials :

      ✯ It is a polynomial of degree 2

      ✯ General form :

                ax² + bx + c  = 0

                   $\boxed{\bf x =\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$

                 

      ✯ 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

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Given,

two zeroes of a polynomial are 1/4 and -1

⇒ Sum of zeroes

                       $=\frac{1}{4} +(-1) \\\\ =\frac{1}{4} -1 \\\\ =\frac{1-4}{4} \\\\ =\frac{-3}{4}$

⇒ Product of zeroes

                       $=(\frac{1}{4})(-1) \\\\ =\frac{-1}{4}$

The quadratic polynomial is of the form :

k[ x² - (sum of zeroes)x + (product of zeroes) ]

$k[x^{2} -(\frac{-3}{4} )x+(\frac{-1}{4})] \\\\ k[x^{2} +\frac{3}{4} x-\frac{1}{4} ] \\\\\\ Put \ k = 4,\\\\ 4[x^{2} +\frac{3}{4} x-\frac{1}{4}] \\\\ 4x^{2} +3x-1$

( substitute any other values in place of k to get many polynomials )