Explain two ways that you can solve for R in the following division statement. Then, use one of your methods to find R.
x^3+4x^2-3x=(x-1)(x^2+5x+2)+R
could you divide x^3+4x^2-3x by (x-1) to get the quotion of (x^2+5x+2)+ the remainder which would be r?
Also what would be another method to solve this?
that is one way.
The other way is to use the Remainder Theorem and evaluate f(1).
That is the remainder R.
Thanks!
Yes, you can divide x^3+4x^2-3x by (x-1) to find the quotient of (x^2+5x+2) and the remainder, which would be R. This can be done using long division or synthetic division.
Method 1: Long Division
To divide x^3+4x^2-3x by (x-1), follow these steps:
1. Write the dividend (x^3+4x^2-3x) and the divisor (x-1) in long division form:
_________
x-1 | x^3 + 4x^2 - 3x
2. Divide the first term of the dividend (x^3) by the first term of the divisor (x). The result is x^2, which becomes the first term of the quotient.
3. Multiply the divisor (x-1) by x^2 and subtract it from the dividend. Write the result below the line:
x^3 + 4x^2 - 3x
- (x^3 - x^2)
5x^2 - 3x
4. Bring down the next term of the dividend (-3x) and repeat steps 2 and 3 until the dividend is fully divisible or the degree of the leftover term is less than the divisor.
5. Continuing the steps, divide 5x^2 by x to get 5x as the next term of the quotient.
6. Multiply the divisor (x-1) by 5x and subtract it from the remaining term:
5x^2 - 3x
- (5x^2 - 5x)
2x
7. The leftover term is 2x, which has a degree lower than the divisor, x-1. Hence, the division is complete.
Therefore, the quotient is x^2+5x+2, and the remainder, R, is 2x.
Method 2: Synthetic Division
Synthetic division is another method used specifically for dividing by linear divisors. Here's how to use it:
1. Write the coefficients of the dividend in decreasing order and exclude the variable x:
1 4 -3 0
2. Write the constant term of the divisor (x-1) on the left side of the division symbol and set up the synthetic division tableau:
1 | 1 4 -3 0
-1 |
3. Bring down the first coefficient (1) in line with the division symbol.
4. Multiply the number outside the division symbol (-1) by the number in the first row (1) and write the result (1) below the second coefficient (4).
5. Add the numbers in the second column (-1 + 1 = 0) and write the result below the line in the third column.
6. Repeat steps 4 and 5 until the entire dividend is processed:
1 | 1 4 -3 0
-1 | -1 -3
-1
7. The result in the last column is the same as the remainder in long division. Therefore, R = -1.
In summary, using long division, the remainder R is 2x, and using synthetic division, the remainder R is -1.