# Determine the remainder of 4x^3 + 5x^2 - 3x + 9 divided by x - 4 and prove the same using long division.

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The remainder when the expression P(x) is divided by x - a is given by P(a).

In the problem, P(x) = 4x^3 + 5x^2 - 3x + 9 and a = 4. The remainder when 4x^3 + 5x^2 - 3x + 9 is divided by x - 4 is 4*4^3 + 5*4^2 - 3*4 + 9 = 333

For verification of the same using long division proceed as follows:

x - 4 | 4x^3 + 5x^2 - 3x + 9 | 4x^2 + 21x + 81

...........4x^3 - 16x^2

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.....................21x^2 - 3x + 9

.....................21x^2 - 84x

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..................................81x + 9

..................................81x - 324

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.............................................333

The reminder of 4x^3 + 5x^2 - 3x + 9 divided by x - 4 is 333 both using long division as well by the use of the remainder theorem.

The remainder is 333.By long division method,we can prove this.

x-4|4x^3+5x^2-3x+9|4x^2+21x+81

4x^3-16x^2

-------------

21x^2-3x+9

21x^2-84x

-------------

81x+9

81x-324

---------------

333

Thus proved

Please refer to the links below regarding how to solve polynomial fractions.

I have checked the workings above with mine, the answer is indeed 333.

**Sources:**

First you have to think what you have to multiply by x to get a 4x^3. To get 4x^3, you'd multiply x by 4x^2. Write that over your long division sign. You also have to multiply this by negative 4. Once you multiply you will subtract your results from the original equation. Continue doing this until you can't anymore. I think this will be easier to understand with a picture, so I'll show my work below. In the end your remainder is **333**.

First set the problem up in long division style, with the 4x^3+5x^2-3x+9 under the half box. Divide the first term of the function being divided (4x^3) by the first term in the function doing the dividing (x), the result is the first term of your answer, which goes above the half box. Then multiply that product out with the rest of the function doing the dividing as if your distributing, put the results under the function under the half box and subtract. Continue on this pattern until you can no longer divide out. This leaves you with a remainder of 333, as shown in the image.

**Sources:**