ANSWER :- x² + 4x + 1

Solution :-

Given :-

⇒ Sum of zeroes, ɑ + β = -4

⇒ Product of zeroes, ɑβ = 1

To Find :-

⇒ Polynomial p(x) satisfying the above condition.

Solution :-

We KNOW,

A standard quadratic polynomial is of the form x² - (sum of zeroes)x + (product of zeroes)

⇒ p(x) = x² - (sum of zeroes)x + (product of zeroes)

⇒ p(x) = x² - (-4)x + (1)

⇒ p(x) = x² + 4x + 1

Hence, The REQUIRED polynomial is x² + 4x + 1

EXTRA INFORMATION :-

◉ The standard form of quadratic polynomial is ax² + bx + c, where a & b are integers and a ≠ 0, c is a constant.

◉ A quadratic polynomial can also be thought as a quadratic equation, So:

• The discriminant of the quadratic polynomial, D = b² - 4ac, If:

⇒ D = 0

Zeroes are equal and real.

⇒ D > 0

Zeroes are real and distinct.

⇒ D < 0

Zeroes are imaginary.

• Its zeroes can also be found in the same way we find of quadratic equation using the Quadratic formula.

Zeroes and its Relationship with the Coefficient :-

• Sum of zeroes :-

Given a quadratic polynomial, p(x) = ax² + bx + c

Then, the sum of zeroes = -b/a

• Product of Zeroes :-

Given a quadratic polynomial, p(x) = ax² + bx + c

Then, the product of zeroes = c/a

A Quick Tip:

If you are given sum of zeroes and product of zeroes then you can find the two zeroes by using the following algorithm:-

• Calculate the difference of zeroes by using the following formula :-

⇒ (ɑ - β)² = (ɑ + β)² - 4ɑβ

• Solve the pair of LINEAR equations, ɑ + β, and ɑ - β, To get the value of zeroes (i.e., ɑ and β)