Remainder Theorem

Prepared by:

Teresita M. Joson

BSEd IV – Math

The Remainder Theorem -is a useful mathematical

theorem that can be used to factorize polynomials of any degree in a neat and fast manner.

The Remainder Theorem states that when you divide a

polynomial P(x) by any factor

(x - a); which is not necessarily a factor of the polynomial; you'll obtain a new smaller polynomial

and a remainder, and this remainder is the value

of P(x) at x = a

Remainder Theorem operates on the fact that a polynomial is

completely divisible once by its factor to obtain a smaller

polynomial and a remainder of zero. This provides an easy way to test whether a value a is a root of the polynomial P(x).

There are two(2) ways to solve for the remainder: Long division/Evaluation

Synthetic division

Using Evaluation:f (x) = x6 + 5x5 + 5x4 + 5x3 + 2x2 – 10x – 8

x-2

In using evaluation method, we first find the value of X, we will use the divisor x-2.

x-2 =0 transposition method

x = 2 + 0

x = 2 you have the value of x.

Next step:

Substitute the vale of x in the equation

F(x) = x6 + 5x5 + 5x4 + 5x3 + 2x2 – 10x – 8

= (2)6 + 5(2)5 + 5(2)4 + 5(2)3 + 2(2)2 – 10(2) – 8

= 64 + 5 (32) + 5 (16) + 5 (8) + 2(4) – 20 – 8

= 64 + 160 + 80 + 40 + 8 – 20 – 8

= 352 – 28

= 324 Remainder

Synthetic Division

Rules:

see to it that the exponents are arranged in descending order

get the numerical coefficients

Using Synthetic Division

Using the same equation:

f (x) = x6 + 5x5 + 5x4 + 5x3 + 2x2 – 10x – 8

x-2

We have the value of x = 2

Arrange the value of numerical coefficient:

Steps:1. Bring down the first value.2. Multiply 1 to your divisor/x.3. Put it down the second value.4. Then add.5. Repeat step 2 until you get the last value.

x=2 1 5 5 5 2 -10 -8Bring

Down 2 14 38 86 176 332

1 7 19 43 88 166 324 remainder

The quotient now in this equation

f (x) = x6 + 5x5 + 5x4 + 5x3 + 2x2 – 10x – 8x-2

• You have to less 1 to your first exponent.

f (x) = x5 + 7x4 + 19x3 + 43x2 + 88x + 166

remainder is 324x-2

Differences:

Evaluation method:You can not determine the quotient.

Synthetic Division:You can get the final quotient of the

equation.

Boardwork!

f (x) = x3 – 4x2 + x – 6 / x + 4

Using evaluation:

x + 4 = 0 x = -4

Substitute:

f (x) = (-4)3 – 4(-4)2 + (-4)– 6

= -64 – 4(16) – 4 – 6

= -64 – 64 – 4 – 6

= -138 remainder

Using Synthetic Division:

f (x) = x3 – 4x2 + x – 6 / x + 4

-4 1 -4 1 -6

-4 32 -132

1 -8 33 -138

We have come up with the same remainder.

Now quotient is:f (x) = x2 - 8x + 33 -138

x + 4

Now! Try this!

Find the remainder using evaluation and Synthetic Division:

1. - 2a5 + 10a4 + a3 + 2a2 – 10a – 3 / a +3

2. 2x4 - 15x3 + 11x2 – 10x – 12 / x – 5

3. x3 + 5x2 – 7x + 2 / x + 2

4. 2x3 - 3x2 – x + 4 / x – 3

5. 5x6 + 2x4 + 5x3 – 5x – 12 / x + 2

Sometimes the questions are complicated

yet the answers are simple…

Thank You!!!