Suppose you divide a polynomial by a binomial. How do you know if the binomial is a factor of the polynomial? Create a sample problem that has a binomial which IS a factor of the polynomial being divided, and another problem that has a binomial which is not a factor of the polynomial being divided.
Can someone give me an example of whats above plz. This is to help me study for a test that is coming up.
To be a factor, the division has to be exact, that is,
the remainder must be zero
works that way with numbers, works that way in algebra.
Sure! I can help you understand how to determine if a binomial is a factor of a polynomial and provide sample problems to practice.
To determine if a binomial is a factor of a polynomial, you can use the method of synthetic division. Here's how it works:
1. Write the polynomial in descending order of powers.
For example, if we have the polynomial: 3x^3 + 2x^2 - 5x - 4, write it as: 3x^3 + 2x^2 - 5x - 4.
2. Write the binomial in the form (x - a), where a is the constant value.
For example, if the binomial is (x - 2), write it as (x - 2).
3. Set up the synthetic division table by writing the coefficients of the polynomial in the first row and the constant term of the binomial (the opposite of a) in the second row.
In our example, the setup would look like:
3 2 -5 -4
|___________
-2
4. Perform synthetic division by bringing down the first coefficient (3) and multiplying it with the constant term (-2), then adding it to the next coefficient (2), and repeating this process until you reach the last coefficient (-4).
The synthetic division steps are as follows:
3 2 -5 -4
|___________
-2
___________
3 -4 3
0 -4 8
5. Analyze the result:
- If the remainder is zero, the binomial is a factor of the polynomial.
- If the remainder is not zero, the binomial is not a factor of the polynomial.
Let's now create two sample problems, one where the binomial is a factor and another where it is not.
1. Example with binomial as a factor:
Polynomial: 2x^3 - 5x^2 - 3x + 6
Binomial: x - 2
Perform synthetic division:
2 -5 -3 6
|___________
2
__________
-3 -9
----------
3 -3
Since the remainder is zero (3 - (-3) = 0), the binomial (x - 2) is a factor of the polynomial.
2. Example with binomial not as a factor:
Polynomial: 4x^3 + 3x^2 - 5x - 2
Binomial: x + 1
Perform synthetic division:
4 3 -5 -2
|___________
-1
__________
4 -1 6
----------
-7
Since the remainder is not zero (-7), the binomial (x + 1) is not a factor of the polynomial.
I hope this helps! Let me know if you have any further questions.