Pre – Calculus Math 12 The Remainder Theorem

Lesson Focus: To divide polynomials by binomials of the form x - a using long division or synthetic division;

to use the remainder theorem to find the remainder from polynomial division

to divide a polynomial by a binomial, use a long division process like the one used with numbers

remember that we multiply then subtract when we do long division

e.g. 7 895

each of the places in long division has a certain name

i.e.

remainder

quotient

dividenddivisor

when we divide polynomials, the following statements are true:

Remainder QuotientDivisor Dividend OR

ax

RxQ

ax

xP

rewrite the previous numerical example in both ways

when we use long division, we are always choosing a term in the quotient that when multiplied to the

leading term of the divisor will produce the leading term of the dividend

e.g. 32832 xxx e.g. 36116 23 mmmm

synthetic division is a shortcut for dividing polynomials by divisors of the form ax , where x is a

variable and a is a constant

when we use synthetic division, we use only the coefficients of the polynomial inside the “L”

set your divisor equal to zero to determine which number goes outside the “L”

e.g. 32832 xxx e.g. 36116 23 mmmm

remember when reading the numbers at the bottom of the synthetic division, you must read them from right

to left

the first number on the right is ALWAYS the remainder

each number to the left is one degree larger than the last number

always make sure that there are not any “holes” in the polynomials before you divide them

e.g. Divide 2by 1168 3 xxx using synthetic division.

remember that function notation is based on SUBSTITUTION

e.g. Find 523 if 2 23 xxxxff .

the remainder theorem states that when a polynomial xP is divided by ax , then the remainder is

aP

we must first set the divisor EQUAL TO ZERO in order to know what to substitute into the function

e.g. Find the remainder when 2by divided is 253 23 xxxx .

a) long division b) synthetic division

c) remainder theorem

we can also determine the value of the coefficients required to give a certain remainder if a particular divisor

is used

e.g. Find k if the remainder on dividing 6 is4by 423 xkxkxx .