Question

find the quadratic polynomial whose zero are 2 and -6. verify the relation between the coefficient and zero of the polynomial​

Answer 1

Answer- The above question is from the chapter 'Polynomials'.

Let's know about quadratic polynomial first.

Quadratic polynomial- A polynomial whose highest power of variable is 2 is called a quadratic polynomial.

Examples:

1) x² + 2x + 2

2) 2x² + 4x + 1

Relationship between zeroes and coefficients of a quadratic polynomial:

Let p(x)= ax² + bx + c be any quadratic polynomial in x.

Let α and β be its zeroes.

Sum of zeroes i.e α and β= -b/a

Product of zeroes i.e αβ= c/a

Given question: Find the quadratic polynomial whose zero are 2 and -6. Verify the relation between the coefficients and zeroes of the polynomial​.

Solution: Let p(x)= ax² + bx + c be any quadratic polynomial in x.

Let α and β be its zeroes.

⇒ α= 2 and β= -6

α + β= -b/a

2 + (-6) = -b/a

-4 = -b/a

4/1 = b/a

αβ= c/a

2 × -6 = c/a

-12/1 = c/a

⇒ a = 1, b = 4 and c = -12

∴ p(x)= x² + 4x - 12 is required quadratic polynomial.

Verification:

Sum of zeroes = 2 + (-6) = -4

Sum of zeroes = -b/a = -4/1

So, sum of zeroes = -b/a

Product of zeroes = 2 × -6 = -12

Product of zeroes = c/a = -12/1 = -12

So, product of zeroes = c/a

Hence, verified.

Answer 2

Given :

• Zeros of quadratic polynomial = 2 & -6

To Find :

• Quadratic polynomial & verify relations b/w zeros & coefficients.

Solution :

Here, zeros of quadratic polynomial : 2 & -6

First finding sum of zeros :

⇒ Sum of zeros = α + ß

⇒ Sum of zeros = 2 + (-6)

⇒ Sum of zeros = 2 - 6

⇒ Sum of zeros = -4

Now finding product of zeros :

⇒ Product of zeros = α × ß

⇒ Product of zeros = 2 × (-6)

⇒ Product of zeros = -12

Now we know,

☛ Quadratic polynomial = x² - (α + ß)x + αß

Putting values :

⇒ Quadratic polynomial = x² - (-4)x + (-12)

⇒Quadratic polynomial = x² + 4x - 12

_______________________

Verification :

Here comparing the polynomial with ax² + bx + c, we get :

• a = 1
• b = 4
• c = -12

Relation 1 :

☛ Sum of zeros = -b/a

⇒ 2 + (-6) = -4/1

⇒ 2 - 6 = -4

⇒ -4 = -4 ⚫Hence verified!⚫

Relation 2 :

☛ Product of zeros = c/a

⇒ 2 × (-6) = -12/1

⇒ -12 = -12 ⚫Hence verified!⚫

Therefore,

Relations b/w zeros & coefficients are verified.