LESSON X
LESSON 6 LONG DIVISION AND SYNTHETIC DIVISION
Example Find [pic].
[pic]
NOTE: [pic]
[pic]
[pic]
The expression [pic] is called the divisor in the division. The function [pic] is called the divisor function.
The expression [pic] is called the dividend in the division. The function [pic] is called the dividend function.
The expression [pic] is called the quotient in the division. The function [pic] is called the quotient function.
The expression [pic] is called the remainder in the division. The function [pic] is called the remainder function.
We have that
[pic]
Multiplying both sides of this equation by [pic], we have that
[pic] =
[pic]
Let a and b be polynomials. Then [pic]. The degree of the remainder polynomial r is less than the degree of divisor polynomial b, written deg r < deg b.
Multiplying both sides of the equation [pic] by [pic], we have that [pic].
Example Find [pic].
[pic]
NOTE: [pic]
[pic]
[pic]
[pic]
The quotient function is [pic] and the remainder function is [pic]. We have that
[pic].
Example Find [pic].
[pic]
The quotient function is [pic] and the remainder function is [pic]. We have that
[pic].
NOTE: In the example above, we had that
[pic]. Thus,
[pic] =
[pic] =
[pic]
Example Find [pic].
[pic]
The quotient function is [pic] and the remainder function is [pic]. We have that
[pic].
Consider the following.
[pic] [pic]
What do the numbers in the third row represent?
[pic] [pic]
Thus, the quotient function is [pic] and the remainder function is [pic]. These are the same answers that we obtained above using long division.
This process is called synthetic division. Synthetic division can only be used to divide a polynomial by another polynomial of degree one with a leading coefficient of one. Thus, you can NOT use synthetic division to find
[pic]. However, we can do the following division.
Example Use synthetic division to find [pic].
[pic] [pic]
Thus, the quotient function is [pic] and the
remainder function is [pic]. These are the same answers that we obtained above using long division.
Example If [pic], then find [pic].
[pic] =
[pic] =
[pic]
This calculation would have been faster (and easier) using the fact that
[pic]
that we obtained in the example above. Thus, [pic], where [pic].
Thus, [pic].
This result can be explained by the following theorem.
Theorem (The Remainder Theorem) Let p be a polynomial. If [pic] is divided by [pic], then the remainder is [pic].
Proof If [pic] is divided by [pic], then [pic]. Thus, [pic].
Example If [pic], then find [pic].
Using synthetic division to find [pic], we have that
[pic] [pic]
Thus, the remainder is [pic]. Thus, [pic] = [pic].
Example If [pic], then find [pic].
Using synthetic division to find [pic], we have that
[pic] [pic]
Thus, the remainder is 41. Thus, [pic].
Theorem (The Factor Theorem) Let p be a polynomial. The expression [pic] is a factor of [pic] if and only if [pic].
Proof [pic] Suppose that [pic] is a factor of [pic]. Then the remainder upon division by [pic] must be zero. By the Remainder Theorem, [pic].
[pic] Suppose that [pic]. By the Remainder Theorem, we have that
[pic]. Thus, [pic]. Thus, [pic] is a factor of [pic].
Example Show that [pic] is a factor of [pic].
We will use the Factor Theorem and show that [pic]. We will use the Remainder Theorem and synthetic division to find [pic].
[pic] [pic]
Thus, [pic]. Thus, by the Factor Theorem, [pic] is a factor of
[pic].
NOTE: The third row in the synthetic division gives us the coefficients of the other factor starting with [pic]. Thus, the other factor is [pic].
Thus, we have that [pic] = [pic].
Example Show that [pic] is not a factor of [pic].
We will use the Factor Theorem and show that [pic]. We will use the Remainder Theorem and synthetic division to find [pic].
[pic] [pic]
Thus, [pic]. Thus, by the Factor Theorem, [pic] is not a factor of
[pic].
Example Find the value(s) of c so that [pic] is a factor of
[pic].
By the Factor Theorem, [pic] is a factor of the polynomial f if and only if [pic].
[pic]
Thus, [pic].
Using the Remainder Theorem and synthetic division to find [pic], we have
[pic] [pic]
By the Remainder Theorem, [pic]. Thus, [pic]
[pic].
Answer: [pic]
Example Find the value(s) of c so that [pic] is a factor of
[pic].
By the Factor Theorem, [pic] is a factor of the polynomial g if and only if [pic].
[pic]
Thus, [pic].
Using the Remainder Theorem and synthetic division to find [pic], we have
[pic] [pic]
By the Remainder Theorem, [pic]. Thus, [pic]
[pic].
Answer: 8
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