# Polynomials

Assume that a1a2a3a4,â€¦are real numbers and x is a real variable. Then f(x) = a1xn + a2xn-1 + a3xn-2 +â€¦ an-1x + anx is called a polynomial.

For example, 5x5 + 3x4 +â€¦+ x is a polynomial in x, where x is a real variable.

# Degree of a Polynomial

Polynomial f(x) = a1xn + a2xn-1 + a3xn-2 +â€¦+ an-1x + anx is a polynomial of degree n, where a1 â‰  0.

For example, 5x5 + 3x4 +â€¦+ x is a polynomial of degree 5.
2x4 + x3 + 4x2 + 2x + 10 is a polynomial of degree 4.
4x3 + 4x2 + 2x + 10 is a polynomial of degree 3.
4x2 + 2x + 10 is a polynomial of degree 2.

Remember that
• The degree of a polynomial is defined for real as well as complex polynomials.
• The degree of a polynomial cannot be a fraction.

# Polynomial Equation

If f(x) is a real or complex polynomial, then f(x) = 0 is the corresponding equation. Simply put, when we equate any polynomial with zero, then it becomes equation.

For example, 5x5 + 3x4 +â€¦+ x = 0 is an equation.
2x4 + x3 + 4x2 + 2x + 10 = 0 is an equation.

# Polynomial Inequation

If f(x) is a real or complex polynomial, then f(x) â‰  0 is the corresponding inequation. Simply put, if any polynomial is not equal to zero, i.e., either greater than zero or less than zero, then it is known as an inequation.

For example, 2x4 + x3 + 4x2 + 2x + 10 >0 is an inequation.
4x2 + 2x + 10 < 0 is an inequation.

# Roots of an Equation

All the values of a variable satisfying the equation are known as the roots of the equation. It is known that an equation of degree n will have n roots, real or imaginary. Roots can be the same or distinct. When the roots are the same, they are known as repeated roots.

Any equation of degree 2 is known as a quadratic equation.

ax2 + bx + c = 0 is known to be the standard equation of quadratic equation. This equation will have two roots either real or imaginary.

# Geometrical Meaning of Roots

For any given equation y = f(x) = 0, the number of times the graph of this equation cuts X-axis is equal to the distinct real roots of this equation.

For example, y = (xâ€“1) (x+2) (xâ€“2)=0 will intersect X-axis at three distinct points, namely x = 1, âˆ’2 and 2.

For any quadratic equation f(x) = (x â€“1) (x âˆ’3), the graph of this equation (as can be seen below) will intersect X-axis at two distinct points, namely x = 1 and x = 3.

So, the equation f(x) = x2 â€“ 4x + 3 = 0 will have two roots, i.e., two values of x = 1 and x =3, which will satisfy this equation.

If we say that the two roots â€˜pâ€™ and â€˜qâ€™ of the quadratic equation f(x) = ax2 + bx + c = 0 are such that f(2) Ã— f(âˆ’2) >0, then either both the roots â€˜pâ€™ and â€˜qâ€™ of this equation will lie inside â€“2 and 2 or both the roots â€˜pâ€™ and â€˜qâ€™ will lie outside the range of â€“2 and 2.

Similarly, if we say that the two roots â€˜pâ€™ and â€˜qâ€™ of the quadratic equation f(x) = ax2 + bx + c = 0 are such that f(2) Ã— f(âˆ’2) <0, then one of the two roots of this equation will lie inside this range and another will lie outside the range of â€“2 and 2.