Find the polynomials given two zeroes 1/4 -1

1.	Find the polynomials given two zeroes 1/4 -1
Answer» 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 _________________________________ 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 )

Answer»

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 )