Question

If α,β are the zeros of the polynomial, x^2+px +3 and α^2 + β^2 = 10, then what is the value of p? *

Answer 1

Answér :

p = ± 4

Note:

★ The possible values of the variable for which the polynomial becomes zero are called its zeros .

★ A quadratic polynomial can have atmost two zeros .

★ The general form of a quadratic polynomial is given as ; ax² + bx + c .

★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;

• Sum of zeros , (α + ß) = -b/a

• Product of zeros , (αß) = c/a

★ If α and ß are the zeros of a quadratic polynomial , then that quadratic polynomial is given as :

k•[ x² - (α + ß)x + αß ] , k ≠ 0.

★ The discriminant , D of the quadratic polynomial ax² + bx + c is given by ;

D = b² - 4ac

★ If D = 0 , then the zeros are real and equal .

★ If D > 0 , then the zeros are real and distinct .

★ If D < 0 , then the zeros are unreal (imaginary) .

Solution :

Here ,

The given quadratic polynomial is ;

x² + px + 3 .

Now ,

Comparing the given quadratic polynomial with the general quadratic polynomial ax² + bx + c ,

we have ;

a = 1

b = p

c = 3

Also ,

It is given that , α and ß are the zeros of the given quadratic polynomial .

Thus ,

=> Sum of zeros = -b/a

=> α + ß = -p/1

=> α + ß = -p ----------(1)

Also ,

=> Product of zeros = c/a

=> αß = 3/1

=> αß = 3 -----------(2)

Also ,

It is given that ,

α² + ß² = 10 ------------(3)

Now ,

We know that , (A + B)² = A² + B² + 2AB

Thus ,

=> (α + ß)² = α² + ß² + 2αß

=> ( p )² = 10 + 2×3

=> p² = 10 + 6

=> p² = 16

=> p = √16

=> p = ± 4

Hence ,

The required answer is p = ± 4 .