How to memorise relationship between zeroes and coefficients of quadratic, cubic quatric polynomials?

Path to memorise the relationship between zeroes and coefficients quadratic & cubic polynomials.

Do you want to verify the relationship between zeroes and coefficients of Quadratic and Cubic Polynomials? Do you find an easy way to learn the relation of zeroes and coefficients? Do you find the questions and solutions based on the verifications of the relationship between zeroes and coefficients?

Then you are at the right place.

In this article, I will guide you, on how to verify the relationship. Some suitable examples based on it. Our team made it easy to learn very soon. The solutions of examples clear all doubts regarding this topic. Do you have any other doubts regarding verification and solutions? You should consult with our teams at any time. the graph on the Polynomials

What do you understand by Quadratic Polynomials?

Quadratic has been derived from 'Quadrate' which means 'square'. The Exponents in polynomials, with the highest degree of 2  are known as Quadratic polynomials. The polynomials in ax²+bx+c, (where a≠0)  are called Quadratic polynomials.

Zeroes of the Quadratic Polynomials: Definition

The values of x of f(x) at which the value of f(x) = 0. The values of x at which the quadratic polynomials become zero, that values of x are known as zeros of Quadratic polynomials. 

When you factorise the quadratic polynomials and obtain the value of x, which are zeros of Quadratic polynomials.

Coefficient of Quadratic Polynomial:

In the term of Quadratic Polynomial, any of the factors with the sign of the term is called the Coefficient of the product of the other factors.

Example:

f(x) = 3x²-5x+4

In f(x), the coefficient of x² is +3.

In f(x), the coefficient of x is -5.

Other Examples 

In -5x, the coefficient of x is -5.

In +x, the coefficient of x is +1.

In -7x², the coefficient of x² is -7.

In -5yx², the coefficient of x² is -5y and the coefficient of y is -5x².

Relationship between the zeroes and coefficients of a Quadratic Polynomial.

Let's 𝝰 and 𝝱 be the zeroes of Quadratic Polynomial f(x) = ax²+bx+c.

Using the factor theorem,

( x-𝜶) and (x-𝝱) are factors of quadratic polynomial f(x).

∴    f(x) = k( x-𝜶)(x-𝝱),  where k is a constant.

⟹ax²+bx+c = k{x²-(𝝰 +𝝱)x+𝝰𝝱}

⟹ax²+bx+c = kx²-k(𝝰 +𝝱)x+k𝝰𝝱

On comparing the coefficients of x², x and constant terms on both sides, we get

Verify the relationship between the zeroes and coefficients of a Quadratic Polynomial.

Let's Quadratic Polynomial f(x) = x²+3x+2.

Zeroes using factorisation.

⟹f(x) = x²+3x+2

⟹f(x) = x²+2x+1x+2

⟹f(x) = x(x+2)+1(x+2)

⟹f(x) = (x+2)(x+1)

Find the Quadratic Polynomial if its zeroes are 𝜶 = 2 and 𝝱 = 3. 

What do you understand by Cubic Polynomial?

Cubic has been derived from 'Cube' which means 'power three'. The Exponents in polynomials, with the highest degree of 3  , are called Cubic polynomials. The polynomials in ax³+bx²+bx+c, (where a≠0)  are known as Cubic polynomials.

Zeroes of the Cubic Polynomials: Definition

The values of x of f(x) at which the value of f(x) = 0. The values of x at which the cubic polynomials become zero, that values of x are known as zeros of Cubic polynomials. 

When you factorise the Cubic polynomials and obtain the value of x, which are zeros of Cubic polynomials.

Coefficient of Cubic Polynomial:

In the term of Cubic Polynomial, any of the factors with the sign of the term is called the Coefficient of the product of the other factors.

Relationship between the zeroes and coefficients of a Cubic Polynomial.

Let's 𝜶, 𝝱 and 𝜸 are zeroes of Cubic Polynomial f(x) = ax³+bx²+cx+d where a≠0.

Using the Factor theorem, 

(x-𝜶),(x-𝝱) and (x-𝜸) are factors of cubic Polynomial f(x). Where k is any non-zero real number. 

Verify the relationship between the zeroes and coefficients of a Cubic Polynomial.

Find the Cubic Polynomial if zeros are 𝜶 = 1, 𝝱 = 2, 𝜸 = 5.

What do you understand by Quatric(Bi-Quadratic) Polynomial?

Quatric has been derived from 'Quad' which means 'Power Four'. The Exponents in polynomials, with the highest degree of 4 are known as Quatric polynomials. The polynomials in ax⁴+bx³+bx²+cx+e, (where a≠0)   are called Quatric(Biquadratic) polynomials.

Zeros of the Quatric Polynomials: Definition

The values of x of f(x) at which the value of f(x) = 0. The values of x at which the quatric polynomials become zero, that values of x are known as zeros of quatric polynomials. 

When you factorise quatric polynomials and obtain the value of x, that are zeros of quatric polynomials.

Coefficient of Quatric (Bi-quadratic) Polynomial:

In the term of Quatric Polynomial, any of the factors with the sign of the term is called the Coefficient of the product of the other factors.

Relationship between the zeroes and coefficients of a Quatric(Bi-Quadratic) Polynomial.

Let's 𝜶, 𝝱, 𝜸 and 𝞭 are zeros of Quatric Polynomial f(x) = ax⁴+bx³+cx²+dx+e where a≠0.

Using the Factor theorem, 

(x-𝜶),(x-𝝱), (x-𝜸) and (x-𝞭) are factors of Quatric Polynomial f(x). Where k is any non-zero real number. 

Verify the relationship between the zeroes and coefficients of a Quatric Polynomial.

Find the Quartic polynomial whose 𝛼= -2, 𝛽=3, 𝛾=-5, 𝛿=-1\2.

Conclusion: 

I have shared complete information regarding the relationship between zeros and coefficients of quadratic, cubic, and quartic polynomials in this article. This clears all doubts by verifying the relation between zeros and coefficients. You can contact our experts if you have any other doubts about this topic.