Question

find the quadratic polynomial for the zeroes alpha and bheta given in each case 1/4,-1​

Answer 1

Answer:-

The Required Polynomial Quadratic Polynomial is $\bf 4x^2+3x-1.$

Explanation:-

Given:-

• $\tt\alpha$ And $\tt\beta$ are zeroes of a quadratic polynomial.

• Also value of $\tt\alpha\:\:is\:\:\dfrac{1}{4}$ And of $\tt\beta\:\:is\:\:-1.$

To Find:-

• The quadratic polynomial.

Concept Used:-

Every Quadratic Polynomial is in the form of,

$\\ \bf\mapsto{x}^{2} - (a + b)x + ab.$

Where,

• a and b are the zeroes of the polynomial.

So Here,

$\\ \sf \mapsto a + b.$

$\\ \sf = \alpha + \beta$

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

$\\ \sf = \frac{1 - 4}{4} .$

$\\ \large\boxed{{ \boxed{ \bf = - \frac{3}{4}. }}}$

$\underline{\bf\therefore\: \alpha+\beta = -\dfrac{3}{4}.}$

Similarly,

$\\ \sf \mapsto ab.$

$\\ \sf = \alpha \beta .$

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

$\\ \large\boxed{ \boxed{ \bf = - \frac{ 1}{4} .}}$

$\underline{\bf\therefore\: \alpha\beta = -\dfrac{1}{4}.}$

Therefore,

Required Polynomial,

$\\ \bf\mapsto{x}^{2} - (a + b)x + ab.$

$\\ \sf = {x}^{2} - ( \alpha + \beta ) + \alpha \beta .$

$\\ \sf = {x}^{2} - \bigg( - \dfrac{ 3}{4} \bigg) + \bigg( - \dfrac{1}{4}\bigg).$

$\\ \sf = {x}^{2} + \frac{3 }{4} - \frac{1}{4} .$

$\\ \sf = \dfrac{4 {x}^{2} + 3x - 1 }{4} \: \: \: \rm(taking \: \: lcm).$

• Now we know that when we have to find out the roots of the equation we would have to put it equal to 0, So we can further simplify the above polynomial.

$\\ \sf \mapsto \frac{4 {x}^{2} + 3x - 1 }{4} = 0...(1)$

$\\ \sf \mapsto4{x}^{2} + 3x - 1 = 0 \times 4.$

$\\ \sf \mapsto4 {x}^{2} + 3x - 1 = 0. . .(2)$

• So we can see that eq(1) and eq(2) both are equal to zero.

$\\ \sf \mapsto \frac{4 {x}^{2} + 3x - 1}{4} = 4 {x}^{2} + 3x - 1 = 0.$

And Hence,

The Polynomial,

$\\ \pink\bullet \boxed{ \boxed{ \bf \frac{4 {x}^{2} + 3x - 1 }{4} = 4 {x}^{2} + 3x - 1 .}}$

Therefore,

The Quadratic Required Polynomial Is,

$\\ \large \red{ \boxed{ \orange{ \boxed { \bf = 4 {x}^{2} + 3 - 1.}}}}$

$\bf\therefore$ The Polynomial Is $\bf 4x^2+3x-1.$

Answer 2

${\huge{\red{\underline{4x²+3x-1}}}}$

${\green{\boxed{MARK \:AS \: BRAINLIEST}}}$

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