Relation Between Coefficients and Zeros of a Polynomial: a Linear Polynomial of the Form

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A linear polynomial of the form . If k is the zero of then,

Zero of the polynomial,

Now, consider the quadratic polynomial,

Factorization of can be done by splitting the middle term into two terms such that their product is a multiple of the first term. i.e. multiple of Middle term can be written as,

Zeros of the polynomial will be same as zeros of

Zeros are found by equating the polynomial to zero.

Coefficients and Zeros of A Polynomial

Coefficients and Zeros of a Polynomial

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Therefore, either

gives and gives .

Zeros of are and .

It is observed that, sum of zeros,

Product of the roots, constant term/coefficient of

Let’s take one more example to verify above concept, let

To factories the above polynomial, we have to split the middle term -3x into two terms such that the product of them is a multiple of

Therefore,

can be written as [since

Zeros of P(x) are,

Sum of zeros,

Product of zeros,

In general, if α and β are the zeros of the polynomial then and are the factors of

can be written as,

where k is a constant.

Comparing the coefficients of terms gives,

It gives,

Therefore,

Sum of zeros,

Product of zeros,

Solved Examples:

Example 1: Find a quadratic polynomial whose sum and product of zeros are and .

Let α and β be zeros of polynomial of form

If , then and

Therefore, one quadratic polynomial satisfying the above condition is

Now, consider the cubic polynomial If are zeros of then,

Example 2: Two zeros of the polynomial are and . Find the third zero of

Let the third root be γ,

Comparing polynomial with gives,

Sum of zeros of the cubic polynomial