how to find the roots of a polynomial equation

Roots of polynomials are the solutions for any given polynomial for which we demand to find the value of the unknown variable. If we know the roots, we can evaluate the value of polynomial to zero. An expression of the class a_northward10ⁿ+ a_n-anex^{due north-i}+ …… + a₁x + a₀, where each variable has a constant accompanying it equally its coefficient is called a polynomial of caste 'n' in variable ten. Each variable separated with an improver or subtraction symbol in the expression is improve known as theterm. The degree of the polynomial is defined as the maximum power of the variable of a polynomial.

For example, a linear polynomial of the course ax + b is called a polynomial of degree 1. Similarly, quadratic polynomials and cubic polynomials have a degree of 2 and 3 respectively.

A polynomial with just one term is known as a monomial. A monomial containing only a constant term is said to be a polynomial of zero degrees. A polynomial can account to null value even if the values of the constants are greater than nix. In such cases, we look for the value of variables which fix the value of entire polynomial to zero. These values of a variable are known every bit the roots of polynomials. Sometimes they are as well termed equally zeros of polynomials.

Roots of Polynomials Formula

The polynomials are the expression written in the course of:
a_{due north}xⁿ+a_{due north-1}x^n-1+……+a₁10+a₀

The formula for the root of linear polynomial such as ax + b is

x = -b/a

The general form of a quadratic polynomial is ax²+ bx + c and if we equate this expression to naught, we get a quadratic equation, i.eastward. ax^two+ bx + c = 0.

The roots of quadratic equation, whose degree is two, such as axⁱⁱ+ bx + c = 0 are evaluated using the formula;

x = [-b ± √(b²– 4ac)]/2a

The formulas for college degree polynomials are a bit complicated.

Roots of iii-degree polynomial

To find the roots of the three-degree polynomial we need to factorise the given polynomial equation start and so that nosotros get a linear and quadratic equation. And so, we can easily determine the zeros of the three-degree polynomial. Let us understand with the aid of an instance.

Example: 2x ^three − x ² − 7x + 2

Divide the given polynomial by 10 – 2 since it is one of the factors.

2x ⁱⁱⁱ − x ⁱⁱ − 7x + ii =(x – 2) (2x ² + 3x – 1)

At present nosotros can get the roots of the above polynomial since we have got one linear equation and one quadratic equation for which we know the formula.

Also, read:

Polynomial Partitioning
Polynomial For Class 10
Polynomials Course nine

Finding Roots of Polynomials

Permit us take an case of the polynomial p(x) of degree 1 as given below:

p(ten) = 5x + i

Co-ordinate to the definition of roots of polynomials, 'a' is the root of a polynomial p(ten), if
P(a) = 0.

Thus, in gild to make up one's mind the roots of polynomial p(10), nosotros accept to observe the value of 10 for which p(x) = 0. Now,

5x + 1 = 0

x = -i/5

Hence, '-1/5' is the root of the polynomial p(x).

Questions and Solutions

Example 1: Bank check whether -ii is a root of polynomial 3x³+ 5x^two+ 6x + 4.

Solution: Let the given polynomial be,

p(x) = 3x³+ 5x²+ 6x + 4

Substituting x = -2,

p(-ii) = three(-2)³+ five (-ii)² + 6(-ii) + 4

p(-2) = -24 + 20 – 12 + four = -12

Hither, p(-two) ≠ 0

Therefore, -2 is not a root of the polynomial 3x³+ 5x²+ 6x + 4.

Example two: Find the roots of the polynomial x² + 2x – 15

Solution: Given x² + 2x – 15

By splitting the centre term,

x² + 5x – 3x – 15

= x(x + 5) – iii(x + 5)

= (x – iii) (x + five)

⇒ x = iii or ten =−5

Video Lesson

Condition for Common Roots

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Frequently Asked Questions – FAQs

What are the roots of a polynomial?

Roots of a polynomial refer to the values of a variable for which the given polynomial is equal to zero. If a is the root of the polynomial p(x), then p(a) = 0.

How many roots does a polynomial take?

The number of roots of any polynomial is depended on the degree of that polynomial. Suppose n is the degree of a polynomial p(10), so p(x) has due north number of roots. For example, if n = 2, the number of roots will be 2.