Question

find a quadriatic polynomial,whose zeroes are 4 and -3

Answer 1

Question :–

Find a quadriatic polynomial,whose zeroes are 4 and -3.

Answer :–

• $\boxed{{x}^{2} - x - 12}$

Given :–

• Zeros of the quadratic polynomial = 4 & -3.

To Find :–

• Quadratic polynomial.

Solution :–

Let the quadratic polynomial be,

$\boxed{ {ax}^{2} + bx + c = 0}$

Zeros of the polynomial :– 4 & -3

So,

$(x - 4) \: and \: (x + 3)$ are zeros of the quadratic polynomial.

$\implies (x - 4)(x + 3)$

$\implies {x}^{2} + 3x - 4x - 12$

$\implies \boxed{{x}^{2} - x - 12}$

Hence,

the quadratic polynomial is $\boxed{{x}^{2} - x - 12}$

Answer 2

Answer:

x² - x - 12

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:

• Given : α = 4 , ß = -3
• To find : Quadratic polynomial

We know that ,

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.

Thus ,

Required set of quadratic polynomials is ;

=> k•[ x² - (α + ß)x + αß ] , k ≠ 0

=> k•[ x² - (4 + (-3))x + 4•(-3) ] , k ≠ 0

=> k•[ x² - (4 - 3)x - (4•3) ] , k ≠ 0

=> k•[ x² - x - 12 ] , k ≠ 0

For k = 1 , the required quadratic polynomial will be ;

=> 1•[ x² - x - 12 ]

=> x² - x - 12

Hence ,

The required quadratic polynomial is ;

x² - x - 12 .