Remainder and Factor theorem Important Formulas, Definitions, & Examples:

Overview:

What is the Remainder Theorem:

The remainder theorem is used to find out the remainder of any expression \(f(x)\) by dividing from a divisor \(f(c)\). where \(f(x)\) and \(f(c)\) could be any linear, polynomial equation or any number.

Example of the Remainder theorem in real life:

If we have to distribute 5 bananas to 4 people, then one banana will be left after distribution, and this is the remainder. \(\frac{5}{4} = 1\).

Basic Remainder Theorem:

We can use a simple method to find the remainder of any number or any linear equation.

Let's find the remainder of an expression \(\frac{x \times y \times z}{k}\). When \(x, \ y, \ and \ z\) divided by \(k\) we get \(\frac{x_a \times y_a \times z_a}{k}\), where \(x_a\), \(y_a\), and \(z_a\) are remainders.

Here \(x_a\) is the remainder of \(\frac{x}{k}\), \(y_a\) is the remainder of \(\frac{y}{k}\) and \(z_a\) is the remainder of \(\frac{z}{k}\). We can understand it by taking some examples given below.

What is the Factor Theorem:

While finding the remainder of any expression, if we get zero as a remainder, then it means the divisor \(f(c)\) is one of the factors of dividend \(f(x)\). In another word, we can say if any dividend value \(f(x)\) is completely divisible by the divisor \(f(c)\), then there will be no remainder left or zero remainders.

Example (1): Find the remainder of \(40\) when divided by \(10\)?

Solution: $$ = \frac{40}{10} = 0 $$ Here, \(0\) is the remainder of \(\frac{40}{10}\) or we can say there is no remainder as number 10 is one of the factors of 40.

Remainder Theorem Examples:

Example (2): Find the remainder of \(55\) when divided by \(7\)?

Solution: $$ = \frac{55}{7} = \frac{6}{7} = 6 $$ Here, \(6\) is the remainder of \(\frac{55}{7}\).

Example (3): Find the remainder of \(9 \times 10 \times 11\) when divided by \(4\)?

Solution: $$ = \frac{9 \times 10 \times 11}{4} \\ = \frac{1 \times 2 \times 3}{4}$$ Here, \(1\) is the remainder of \(\frac{9}{4}\), \(2\) is the remainder of \(\frac{10}{4}\), and \(3\) is the remainder of \(\frac{11}{4}\)

and the final remainder is $$ \frac{1 \times 2 \times 3}{4} = \frac{6}{4} = 2 $$

Example (4): Find the remainder of \(25 \times 28 \times 31\) when divided by \(6\)?

Solution: $$ = \frac{25 \times 28 \times 31}{6} \\ = \frac{1 \times 4 \times 1}{6}$$ where, \(1\) is the remainder of \(\frac{25}{6}\), \(4\) is the remainder of \(\frac{28}{6}\), and \(1\) is the remainder of \(\frac{31}{6}\)

and the final remainder is $$ \frac{1 \times 4 \times 1}{6} = \frac{4}{6} = 4 $$

Example (5): Find the remainder of \(43 \times 46 \times 48\) when divided by \(11\)?

Solution: $$ = \frac{43 \times 46 \times 48}{11} \\ = \frac{10 \times 2 \times 4}{11}$$ where, \(10\) is the remainder of \(\frac{43}{11}\), \(2\) is the remainder of \(\frac{46}{11}\), and \(4\) is the remainder of \(\frac{48}{11}\)

and the final remainder is $$ \frac{10 \times 2 \times 4}{11} = \frac{80}{11} = 3 $$

Example (6): Find the remainder of \(x^3\) when divided by \(x\)?

Solution: $$ = \frac{x^3}{x} = 0 $$ Here, \(0\) is the remainder of \(\frac{x^3}{x}\) or we can say there is no remainder as \(x\) is one of the factors of \(x^3\). We can also say that \(x^3\) is completely divisible by \(x\) which is why no remainder left.

Example (7): Find the remainder of \(x\) when divided by \(2x\)?

Solution: $$ = \frac{x}{2x} = \frac{1}{2} = 1 $$ Here, \(1\) is the remainder of \(\frac{x}{2x}\).

We have discussed the polynomial remainder theorem in the next chapter.