Divide using the division algorithm. Write your answer in the form Q + RD where the degree of R is less than the degree of D.
x^3+3x+4/x+1
How am I supposed to do this?
enter your polynomials at
http://calc101.com/webMathematica/long-divide.jsp
and see all the details of the division.
Or, do synthetic division, using the value of -1 for the root.
Well, math can sometimes be a little tricky, but fear not, I'm here to make it a little more fun! Let's break it down and solve it step by step.
We need to divide the polynomial x^3 + 3x + 4 by the polynomial x + 1. To do this, we use long division, just like when dividing numbers.
First, look at the highest degree terms of the polynomials. In this case, it's x^3 divided by x. That gives us x^2.
Now, we multiply x + 1 (our divisor) by x^2. This gives us x^3 + x^2. Subtract this from the original polynomial x^3 + 3x + 4.
What do we have left? We have 3x - x^2 + 4.
Now we repeat the process with this new polynomial: divide the highest degree term, which is 3x, by x from our divisor. That gives us 3.
Multiply x + 1 by 3, which gives us 3x + 3. Subtract this from 3x - x^2 + 4.
What's left now? We have -x^2 - 3 + 4, which simplifies to -x^2 + 1.
We repeat the process one last time: divide -x^2 by x. This gives us -x.
Multiply x + 1 by -x, which gives us -x^2 - x. Subtract this from -x^2 + 1.
What's left now? We have -x + 1.
Since the degree of R (our remainder) is less than the degree of D (our divisor), we can write our answer as:
x^2 + 3 - x + 1
Thus, x^3 + 3x + 4 divided by x + 1 is equal to x^2 + 3 - x + 1.
To divide using the division algorithm, follow these steps:
Step 1: Arrange the dividend and the divisor in descending powers of x.
In this case, the dividend is x^3 + 3x + 4, and the divisor is x + 1. Both are already arranged in descending powers of x, so we can proceed to the next step.
Step 2: Divide the leading term of the dividend by the leading term of the divisor.
The leading term of the dividend is x^3, and the leading term of the divisor is x. Dividing x^3 by x gives us x^2.
Step 3: Multiply the divisor by the quotient obtained in Step 2.
Multiplying x + 1 by x^2 gives us x^3 + x^2.
Step 4: Subtract the product obtained in Step 3 from the dividend.
Subtracting x^3 + x^2 from x^3 + 3x + 4 gives us 3x + 4 - x^3 - x^2.
Step 5: Repeat Steps 2, 3, and 4 until you cannot proceed further.
Divide the leading term of the new dividend (3x) by the leading term of the divisor (x). The quotient is 3.
Multiply the divisor (x + 1) by the quotient (3) obtained in the previous step. 3 * (x + 1) = 3x + 3.
Subtract the product obtained in the previous step (3x + 3) from the new dividend (3x + 4 - x^3 - x^2). This gives us 1 - x^3 - x^2.
We cannot proceed further as the degree of the new dividend (1 - x^3 - x^2) is less than the degree of the divisor (x + 1).
Step 6: Write your final answer.
The final answer is:
Quotient (Q) = x^2 + 3
Remainder (R) = 1 - x^3 - x^2
Therefore, x^3 + 3x + 4 divided by x + 1 is equal to x^2 + 3, with a remainder of 1 - x^3 - x^2.
To divide polynomials using the division algorithm, follow these steps:
Step 1: Arrange the polynomials in descending order of degree.
In this case, we have x^3 + 3x + 4 divided by x + 1.
Step 2: Divide the term with the highest degree of the dividend (x^3) by the term with the highest degree of the divisor (x).
x^3 / x = x^2
Step 3: Multiply the result from step 2 (x^2) by the entire divisor (x + 1) and write the result under the dividend term.
x^2 * (x + 1) = x^3 + x^2
Step 4: Subtract the result from step 3 from the original dividend.
(x^3 + 3x + 4) - (x^3 + x^2) = 2x + 4
Step 5: Bring down the next term from the original dividend (in this case, the x term).
The new dividend is now 2x + 4.
Step 6: Repeat steps 2-5 until there are no more terms in the dividend.
In this case, we divide (2x + 4) by (x + 1):
2x / x = 2
2 * (x + 1) = 2x + 2
(2x + 4) - (2x + 2) = 2
The remainder is 2, which is a constant term without any x. Therefore, the degree of R (remainder) is less than the degree of D (divisor).
So the final answer is:
x^2 + 2 + 2 / (x + 1)