Degree of a Polynomial Remainder Theorem

Degree of a Polynomial

The highest value of the exponent of a variable (Polynomial of One Variable only) or the total of highest value of the exponent of the variables (Polynomial of Multivariable) among the terms of a Polynomial is called its Degree.

For example:

5x³ + 25x – 7x²– 6

Here, in this polynomial, we have

2. 5x³yz² + 25xy³z⁶ – 7x²– 6

Here, in this polynomial, we have

Reminder Theorem

If any polynomial p(x) is divided by a polynomial q(x) = x – a, then the reminder is p(a). This is known as Reminder Theorem.

Reminder Theorem is helpful in finding the reminder without performing actual Division of one polynomial by another polynomial.

Let us illustrate this by using one example-

Suppose p(x) = 6x³ + 5x² – 7x + 2 is a polynomial. On dividing this polynomial by another polynomial q(x) = x – 5, the reminder can be calculated without performing actual division as

Let us verify this task by actual division process.

The reminder calculated using reminder theorem = Reminder calculated by performing actual division

Factor Theorem

The Exact Divisor of a number is known as its Factor. If first number is divided by a second number and gives reminder as 0(zero), then the Second number is said to be the exact divisor of the First number and accordingly, the Second number is the factor of the First number. For example, when 6 is divided by 2, we get reminder as 0(zero) and so,

If p(x) is any polynomial, such that p(a) = 0 then x – a is the factor of the polynomial p(x). This is known as Factor Theorem.

If x – a is a factor of the Polynomial p(x), then p(a) = 0. This is the converse of Factor Theorem.

Let us illustrate examples of Factor Theorem:-

Suppose p(x) = 6x³ + 5x² – 7x + 2 is a polynomial. On dividing this polynomial by another polynomial x – 5, then according to Reminder Theorem, the reminder will be

Reminder, p(5) ≠ 0 (zero) and so, x – 5 is not a factor of 6x³ + 5x² – 7x + 2

Let us take another example.

Suppose p(x) = 5x² – 7x + 2 is a polynomial. On dividing this polynomial by another polynomial x – 1 and in accordance with Reminder Theorem, the reminder will be