• Nov 6th 2008, 06:22 AM
14041471
1 Simplify x^2 +4x+3 / x+3

2 simplify x^2 -1 / x+1

3 Divide x^2 - 1 into f(x) = 2x^3 + 3x^2 + 9x + 5. Write your answer in the form given by the Division Algorithm; in other words, write your answer as: 2x^3 + 3x^2 + 9x + 5 = q(x)(x^2 - 1) + r(x) for some polynomials q(x) and r(x).

4. Factor the expression x^3 + 8y^3
• Nov 6th 2008, 07:10 AM
ajj86
Solutions
1 Simplify x^2 +4x+3 / x+3

2 simplify x^2 -1 / x+1

3 Divide x^2 - 1 into f(x) = 2x^3 + 3x^2 + 9x + 5. Write your answer in the form given by the Division Algorithm; in other words, write your answer as: 2x^3 + 3x^2 + 9x + 5 = q(x)(x^2 - 1) + r(x) for some polynomials q(x) and r(x).

4. Factor the expression x^3 + 8y^3

1. Well, for the first one, you can factor the numerator so that it is of the form: (x+3)(x+1). Use FOIL to verify this for reinforcement. What do you get when you divide this by the denominator.

2. The second one has a numerator which is the difference of squares. It is of the form (a + b)(a - b). When you write it of this form you get cancellations in the numerator and denominator.

4. For the last one, this is the sum of 2 cubes. It is written of the form:
(a + b)*(a^2 - ab + b^2). Use FOIL to verify this and then make an attempt to do it for the one above. To find a and b, take the cube root of each term. (What is the cube root of x^3? What is the cube root of 8*y^3?) Then, you just need to find the a^2 and b^2 values to substitute into the above form.
• Nov 6th 2008, 07:35 AM
ajj86
Question 3
3 Divide x^2 - 1 into f(x) = 2x^3 + 3x^2 + 9x + 5. Write your answer in the form given by the Division Algorithm; in other words, write your answer as: 2x^3 + 3x^2 + 9x + 5 = q(x)(x^2 - 1) + r(x) for some polynomials q(x) and r(x).

Okay, for this one you start by looking solely at the x^2 in the x^2-1 term. You ask yourself how many times this can be divided into 2x^3. Well, it would be 2x times since x^2 * 2x = 2x^3.

So, you multiply 2x by (x^2 - 1) and get 2x^3 - 2x. Now, subtract this from 2x^3 + 3x^2 + 9x + 5 to get:

2x^3 + 3x^2 + 9x + 5 - (2x^3 - 2x), which is:
3x^2 + 11x +5

In the next step, we ask ourselves again how many times x^2 goes into 3x^2. Well, it would be 3 since 3*x^2 = 3x^2. So, you multiply 3 by (x^2 - 1) and get 3x^2 - 3. Now, subtract this from 3x^2 + 11x + 5 to get:

3x^2 + 11x + 5 - (3x^2 - 3), which is:
11x + 8. You can't divide x^2 -1 into this any further, so you stop.

Using the given form:2x^3 + 3x^2 + 9x + 5 = q(x)(x^2 - 1) + r(x)
The right hand side is written as:

(2x+3)(x^2-1) + (11x + 8). hope this helps
• Nov 6th 2008, 07:39 AM
14041471
thank you
thanks you ever so much, i have afew other question and would really appreacate it if you awnsered them for me

1 Divide x^2 - 1 into f(x) = 2x^3 + 3x^2 + 9x + 5.

2divide x+1 into 2x^3 + 3x^2 + 9x +5
• Nov 6th 2008, 07:55 AM
ajj86
2divide x+1 into 2x^3 + 3x^2 + 9x +5

The first one should already be up. It took me a little longer to get so I didn't add the solution right away.

So, for this one we follow similar steps.
We start by asking ourselves how many times x can be divided into 2x^3. Well, it would be 2x^2 times since x*2x^3 = 2x^3.

So, we multiply (x+1) by 2x^2 to get 2x^3 + 2x^2. Then, we subtract this from 2x^3 + 3x^2 + 9x +5 to get:

2x^3 + 3x^2 + 9x +5 - (2x^3 + 2x^2), which is:
x^2 + 9x +5.

Next step, we want to know how many times x can be divided into x^2. It is x times, since x*x = x^2.

So, we multiply (x+1) by x to get: x^2 + x. Then, we subtract this from x^2 + 9x +5 to get:

x^2 + 9x +5 - (x^2 + x), which is:
8x + 5.

The final step would be to ask how many times x goes into 8x. Well, it would be 8 times since 8*x = 8x. So, we multiply (x+1) by 8 to get 8x+8. Then, we subtract it from 8x + 5 to get:

8x + 5 - (8x + 8), which is:
-3.

(x+1) cannot be divided into -3 any further, so we stop and write the form:

2x^3 + 3x^2 + 9x +5 = (2x^2 + x + 8)(x+1) -3

Expand out the right hand side of this equation to verify that they are equivalent.
• Nov 6th 2008, 08:07 AM
14041471
is this correct?
You are told that f(x) is a polynomial, and you are told that when we divide x+1 into f(x) then the remainder is 0. Based on this information, which of these statements is necessarily true? (Points :5)
f(1) = 0
f(0) = 1
f(-1) = 0
f(0) = -1
• Nov 6th 2008, 08:37 AM
ajj86
Solution
You are told that f(x) is a polynomial, and you are told that when we divide x+1 into f(x) then the remainder is 0. Based on this information, which of these statements is necessarily true? (Points :5)
f(1) = 0
f(0) = 1
f(-1) = 0
f(0) = -1

f(x) = 2x^3 + 3x^2 + 9x +5

When x+1 was divided into this f(x), it yielded a remainder of -3. So I added 3 to the f(x) so that the remainder would be 0.

It looks like: f(x) = 2x^3 + 3x^2 + 9x +8
So, just try the following:

f(1) = 2*1^3 + 3*1^2 + 9*1 + 8 = 22

f(0) = 2*0^3 + 3*0^2 + 9*0 + 8 = 8

f(-1) = 2*(-1)^3 + 3*(-1)^2 + 9*-1 + 8
f(-1) = -2 + 3 - 9 + 8 = 0

f(-1) seems to be the only one that necessarily holds. Hope this helps