Synthetic Division and the Remainder and Factor Theorems
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Pre-Calculus › Synthetic Division and the Remainder and Factor Theorems
Is a root of
?
No
Yes
Maybe
Explanation
To determine if is a root of the function given, you can use synthetic division to see if it goes in evenly. To set up the division problem, set up the coefficients of the function
and then set 1 outside. Bring down the 1 (of the coefficients. Then multiply that by the
being divided in. Combine the result of that
with the next coefficient
, which is
. Then, multiply that by
. Combine that result
with the next coefficient
, which gives you
. Multiply that by
, which gives you
. Combine that with the last coefficient
, whcih gives you
. Since this is not
, you have a remainder, which means that
does not go in evenly to this function and is not a root.
Is a root of
?
No
Yes
Maybe
Explanation
To determine if is a root of the function given, you can use synthetic division to see if it goes in evenly. To set up the division problem, set up the coefficients of the function
and then set 1 outside. Bring down the 1 (of the coefficients. Then multiply that by the
being divided in. Combine the result of that
with the next coefficient
, which is
. Then, multiply that by
. Combine that result
with the next coefficient
, which gives you
. Multiply that by
, which gives you
. Combine that with the last coefficient
, whcih gives you
. Since this is not
, you have a remainder, which means that
does not go in evenly to this function and is not a root.
Divide the polynomial by
.
Explanation
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We mulitply what's below the line by 1 and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
What is the result when is divided by
?
Explanation
Our first step is to list the coefficiens of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficient.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with reminder
This can be rewritten as:
What is the result when is divided by
?
Explanation
Our first step is to list the coefficiens of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficient.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with reminder
This can be rewritten as:
Divide the polynomial by
.
Explanation
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We mulitply what's below the line by 1 and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
Divide the polynomial by
.
Explanation
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder
This can be rewritten as:
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
Divide the polynomial by
.
Explanation
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder
This can be rewritten as:
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
Divide the polynomial by
.
Explanation
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder
This can be rewritten as
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.
Divide the polynomial by
.
Explanation
Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient.
We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients.
To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder.
with remainder
This can be rewritten as
Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial.