long division with variables!!?

Question

plllease help. i know how to use the synthetic method but just using long division how do i solve this problem?

(x^3=4x^2-3x-12)/(x-3)

Answer 1

I presume that you are comfortable with the method of long division using just numbers ? Division of an expression like x^3 + 4x^2 - 3x - 12 by x - 3 proceeds in essentially the same sort of way. ( I presume that the " = " sign in front of " 4x^2 " is really meant to be a " + " sign ? )

Set out the problem just as you would with numbers :

x - 3 ) x³ + 4x² - 3x - 12

( I cannot draw a long bar over the cubic expression to complete the " long division " symbol, but imagine that it is there ).

Right : now say to yourself " x goes into x³ how many times ? " (or, what amounts to the same thing, "what do I multiply x by in order to get x³ ? " )

The answer of course is " x² " ; so enter " x² " in the answer space, above the 4x² term of the dividend. Now you have to multiply the whole of the divisor (x - 3 ) by x², just as you would using numbers, and put the result below the corresponding terms in the cubic expression. The answer to this part is of course x³ - 3x², which you write below the x³ + 4x² of the dividend.

Now subtract x³ - 3x² from x³ + 4x² to get 7x² ( careful here ! 4x² - ( - 3x² ) = 7x² ! )

Again, as with numbers, you " bring down " the next number, which here is the term " - 3x " ; so the next part is to divide ( x - 3 ) into ( 7x² - 3x ). Say to yourself " x into 7x² goes 7x " . So enter " 7x " in the answer space, above the " - 3x " term.

Now multiply the divisor ( x - 3 ) by 7x to give 7x² - 21x, and subtract it from 7x² - 3x. The remainder this time is 18x : ( - 3x - ( -21x ) ) = ( - 3x + 21x ) = 18x ).

Bring down the last term, - 12, giving 18x - 12 for the last part. " x into 18x goes 18 "; so enter 18 into the answer above the " - 12 " ; multiply ( x - 3 ) by 18 giving 18x - 54 ; write this below the 18x - 12 and subtract, to get the remainder 42.

And there you have it. ( x - 3 ) goes into x³ + 4x² - 3x - 12 ( x² + 7x + 18 ) times, with a remainder of 42.

Verify the answer by multiplying out ( x² + 7x + 18 ) ( x - 3 ), which gives x³ + 4x² - 3x - 54 ; you have to add in the remainder 42 of course, giving x³ + 4x² - 3x - 12, which is what you started with.

It is a bit long-winded to explain of course, but you should soon get the idea with a bit of practice. Hope it helps.

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Answer 2

You have made a mistake in entering the question
Is it - 4x²
OR
+ 4x²
?????