Pre Calculus Math 12 The Remainder Theorem · synthetic division is a shortcut for dividing...
Transcript of Pre Calculus Math 12 The Remainder Theorem · synthetic division is a shortcut for dividing...
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 .