Part 1: 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.
I do not get this at all and help would mean so much
the divisor is a factor if the remainder is zero
Just as you know that 5 is a factor of 35 because 35/5 has zero remainder.
For example,
(2x-1)(x+2)(3x+7) / (x+2)
= (6x^3+23x^2+15x-14)/(x+2)
we know that the remainder is zero, because we constructed it that way. To make that a fraction where the divisor is NOT a factor, just change any of the coefficients. Then the divisor will no longer go in evenly.
To determine if a binomial is a factor of a polynomial, you can use the Remainder Theorem. According to the Remainder Theorem, if you divide a polynomial P(x) by a binomial (x - a), where 'a' is a constant, and the remainder is zero, then (x - a) is a factor of the polynomial. In other words, when the remainder is zero, the binomial is a factor.
Sample Problem 1:
Let's consider the polynomial P(x) = x^3 - 2x^2 - x + 2 and the binomial (x - 1). To check if (x - 1) is a factor, we divide P(x) by (x - 1) using long division:
x^2 + x - 1
_________________________
x - 1 | x^3 - 2x^2 - x + 2
We start by dividing x into x^3, which gives us x^2. Then, we multiply each term of (x - 1) by x^2:
x^2 * (x - 1) = x^3 - x^2
Now, we subtract this result from the polynomial:
x^3 - 2x^2 - (x^3 - x^2) = -x^2 - x^2
Next, we bring down the next term, -x:
_________________________
x^2 + x - 1
x - 1 | x^3 - 2x^2 - x + 2
- x^3 + x^2
_____________
-3x^2 - x
Dividing x into -3x^2 gives us -3x. Multiplying each term of (x - 1) by -3x:
-3x * (x - 1) = -3x^2 + 3x
Subtracting this result from the polynomial gives us:
(-3x^2 - x) - (-3x^2 + 3x) = 2x
Finally, we bring down the last term, 2:
_________________________
x^2 + x - 1
x - 1 | x^3 - 2x^2 - x + 2
- x^3 + x^2
_____________
- 3x^2 - x
+ 3x^2 - 3x
____________
- 4x + 2
The remainder is -4x + 2, which is not zero. Thus, (x - 1) is not a factor of P(x).
Sample Problem 2:
Let's consider the polynomial P(x) = 2x^2 + 3x - 1 and the binomial (x + 1). To check if (x + 1) is a factor, we divide P(x) by (x + 1) using long division:
2x - 1
____________________
x + 1 | 2x^2 + 3x - 1
Dividing x into 2x^2 gives us 2x. Multiplying each term of (x + 1) by 2x:
2x * (x + 1) = 2x^2 + 2x
Now, we subtract this result from the polynomial:
(2x^2 + 3x) - (2x^2 + 2x) = x
Finally, we bring down the last term, -1:
____________________
2x - 1
x + 1 | 2x^2 + 3x - 1
- 2x^2 - 2x
_______________
x - 1
The remainder is x - 1, which is not zero. Therefore, (x + 1) is not a factor of P(x) either.
In summary, when the remainder is zero, the binomial is a factor of the polynomial. If the remainder is not zero, the binomial is not a factor.
Dividing a polynomial by a binomial is known as polynomial long division. To determine if the binomial is a factor of the polynomial, you can follow these steps:
Step 1: Arrange the polynomial and the binomial in descending order.
Step 2: Divide the leading term of the polynomial by the leading term of the binomial.
Step 3: Multiply the result from step 2 by the binomial, and subtract this product from the polynomial.
Step 4: Repeat steps 2 and 3 with the resulting polynomial until you can't perform any more divisions.
Now, let's illustrate with sample problems:
Sample Problem 1: Binomial is a Factor
Consider the polynomial: f(x) = x^3 + 3x^2 - x - 3
And the binomial: g(x) = x + 1
We can check if g(x) is a factor of f(x) by performing polynomial long division:
```
x^2 + 2x - 3
_________________________
x + 1 | x^3 + 3x^2 - x - 3
- (x^3 + x^2)
____________
2x^2 - x
- (2x^2 + 2x)
___________
-3x - 3
- (-3x - 3)
___________
0
```
Since we reach a remainder of 0, we can conclude that (x + 1) is a factor of f(x).
Sample Problem 2: Binomial is Not a Factor
Consider the polynomial: f(x) = x^3 + 2x^2 + x + 2
And the binomial: g(x) = x - 1
Let's perform polynomial long division to check if g(x) is a factor of f(x):
```
x^2 + 3x + 4
_______________________
x - 1 | x^3 + 2x^2 + x + 2
- (x^3 - x^2)
___________
3x^2 + x
- (3x^2 - 3x)
__________
4x + 2
- (4x - 4)
__________
6
```
Since there is a remainder of 6 and we haven't reached 0, we can conclude that (x - 1) is not a factor of f(x).
By following these steps and working through the polynomial long division, you can determine if a binomial is a factor of a polynomial.