Question

if alpha and beta are zeros of the quadratic polynomial 4x square + 4 x + 1 then form a quadratic polynomial whose zeros are 2 alpha and 2 Beta​

Answer 1

Answer :

The required quadratic polynomial is : x² + 2x + 1

Given :

The quadratic polynomial

• 4x² + 4x + 1
• α and ß are the zeroes of the given polynomial

To Find :

• A quadratic polynomial whose zeroes are 2α and 2ß

Concept to be used :

Relationship between the zeroes and the coefficients of a quadratic polynomial

$\sf \bullet \: \: Sum \: of \: the \: zeroes = -\dfrac{Coefficient\: of \: x}{Coefficient \: of \: x^{2}} \\\\ \sf \bullet \: \: Product\: of \: the \: zeroes = \dfrac{Constant \: term}{Coefficient \: of \: x^{2}}$

Solution :

Since α and ß are the zeroes of polynomial ,

4x² + 4x + 1

So from sum relation ,

$\sf \implies \alpha + \beta = -\dfrac{4}{4} \\\\ \sf \implies \alpha + \beta = -1 \\\\ \sf Multiplying \: both \: side \: by \: 2 \\ \sf \implies 2(\alpha + \beta ) = -2 \\\\ \implies 2\alpha + 2\beta = -2$

From product relation ,

$\sf \implies \alpha \beta = \dfrac{1}{4} \\\\ \sf \implies 4\alpha \beta = 1$

Again we know that , the expression of a quadratic polynomial can be given by ,

$\sf \dashrightarrow x^{2} - (sum \: of \: the \: zeroes)x + product \: of \: the \: zeroes \\\\ \sf \dashrightarrow x^{2} -(-2)x + 1 \\\\ \sf \dashrightarrow x^{2} + 2x + 1$