Question

find the quadratic polynomial whose zeroes are 1 and -3. Verufy the relation between the coefficient and the zeroes of the polynomial

Answer 1

Given:

The two zeroes of a quadratic polynomial are (1), (-3)

To find:

The quadratic polynomial whose zeroes are (1), (-3), and also verify the relation between the coefficient and the zeroes of the polynomial.

So,

• To find a quadratic polynomial

k[x² - (sum of zeroes)x + (product of the zeroes)]

⇒ Here 'k' is a 'constant'

Now,

Let zeroes be

• α = 1 and
• β = (-3)

The sum of the zeroes will be

(α+β) = 1 + (-3) = (-2)

And

The product of zeroes will be

αβ = (1) * (-3) = (-3)

Now,

k[x² - (-2)x + (-3)]

= k[ x² + 2x - 3 ]

Hence,

• The required polynomial is x² + 2x - 3

Verification of the coefficients and zeroes.

We know the standard form of the quadratic polynomial

• ax² + bx + c

So, a = 1, b = 2, and c = (-3)

Now,

Sum of the zeroes = (-b/a) = (coefficient of x)/(coefficient of x²) = (-2/1) = (-2)

And

Product of the zeroes = (c/a) = (constant term)/(coefficient of x²) = (-3/1) = (-3)

Answer 2

Answer:

Given:

The two zeroes of a quadratic polynomial are (1), (-3)

To find:

The quadratic polynomial whose zeroes are (1), (-3), and also verify the relation between the coefficient and the zeroes of the polynomial.

So,

To find a quadratic polynomial

k[x² - (sum of zeroes)x + (product of the zeroes)]

⇒ Here 'k' is a 'constant'

Now,

Let zeroes be

α = 1 and

β = (-3)

The sum of the zeroes will be

(α+β) = 1 + (-3) = (-2)

And

The product of zeroes will be

αβ = (1) * (-3) = (-3)

Now,

k[x² - (-2)x + (-3)]

= k[ x² + 2x - 3 ]

Hence,

The required polynomial is x² + 2x - 3

Verification of the coefficients and zeroes.

We know the standard form of the quadratic polynomial

ax² + bx + c

So, a = 1, b = 2, and c = (-3)

Now,

Sum of the zeroes = (-b/a) = (coefficient of x)/(coefficient of x²) = (-2/1) = (-2)

And

Product of the zeroes = (c/a) = (constant term)/(coefficient of x²) = (-3/1) = (-3)