Question

Find the quadratic polynomial whose zeroes are
$(5 + 2 \sqrt{3} )$
and
$(5 - 2 \sqrt{3} )$
​

Answer 1

Question:

Find the quadratic polynomial whose zeros are

(5+2√3) and (5-2√3).

Answer:

x² - 10x + 13

Note:

• A polynomial of degree two is said to be quadratic polynomial.

• A quadratic polynomial can have atmost two zeros.

• The general form of a quadratic polynomial is ; ax² + bx + c .

Also,

If A and B are the zeros of the quadratic polynomial ax² + bx + c , then ;

Sum of zeros (A+B) = -b/a

Product of the zeros (A•B) = c/a

• If A and B are the zeros of any quadratic polynomial , then it will be given as ;

x² - (A+B)x + A•B

Solution:

Here,

The zeros of required quadratic polynomial are ;

(5+2√3) and (5-2√3).

Let ;

A = (5+2√3)

B = (5-2√3)

Now,

The sum of the zeros of required quadratic polynomial will be ;

=> A + B = (5+2√3) + (5-2√3)

=> A + B = 5 + 5 + 2√3 - 2√3

=> A + B = 10 ---------(1)

Also,

The product of the zeros of the required quadratic polynomial will be ;

=> A•B = (5+2√3)•(5-2√3)

=> A•B = (5)² - (2√3)²

=> A•B = 5² - 2²(√3)²

=> A•B = 25 - 4•3

=> A•B = 25 - 12

=> A•B = 13 ----------(2)

Now,

The required quadratic polynomial will be given as ;

=> x² - (A+B)x + A•B

=> x² - 10x + 13 {using eq-(1) and eq-(2)}

Hence,

The required quadratic polynomial is ;

x² - 10x + 13 .

Answer 2

For showing full answer click on photo.

Hope it helps you.