Help with this long division problem?

Question

x^4 + 5x^3 - 30 / x^2 - 3

Find the quotient and remainder

Thanks for the help

Answer 1

x^4 + 5x^3 - 30 / x^2 - 3

The important thing is that, especially with division, polynomials need place value so 0 coefficients apply.

Rewrite as follows

x^4 + 5x^3 + 0x^2 + 0x - 30 / x^2 + 0x - 3

.................... x² + 5x + 3
................. ____________________
x² + 0x - 3 )x^4 + 5x³ + 0x² + 0x - 30
.................. x^4 + 0x³ - 3x²
................. ------------------ .....↓
........................... 5x³ + 3x² + 0x
........................... 5x³ + 0x² - 15x
........................... ------------------ ....↓
.................................... 3x² + 15x - 30
..................... ............... 3x² + 0x - 9
.................................... ------------------
....................... ................... 15x - 21

So x^4 + 5x³ - 30 = (x² - 3)(x² + 5x + 3) + 15x - 21
(Or quotient is x² + 5x + 3 and the remainder is 15x - 21)

Answer 2

Divide leading terms: x^4/x^2 = x^2. Then x^2 is the first term in your quotient.

x^4 + 5x^3 - 30 - x^2(x^2-3) =
x^4 + 5x^3 - 30 - x^4 + 3x^2 =
5x^3 + 3x^2 - 30

Divide this by x^2 - 3, by dividing leading terms: 5x^3 / x^2 = 5x, so 5x is the second term in your quotient.

5x^3 + 3x^2 - 30 - 5x(x^2-3) =
5x^3 + 3x^2 - 30 - 5x^3 + 15x =
3x^2 + 15x - 30

Divide this by x^2 - 3. Divide leading terms: 3x^2/x^2 = 3, so 3 is the third (and last) term in your quotient.

3x^2 + 15x - 30 - 3(x^2-3) =
3x^2 + 15x - 30 - 3x^2 + 9 =
15x - 21

So 15x - 21 is the remainder, and x^2 + 5x + 3 is the quotient.

Answer 3

Always look at the terms with the highest exponent.

x^2-3 ) x^4 - 5x^3 -30

x^2 goes into x^4 x^2 times

x^2(x^2-3) = x^4-3x^2.

Subtract this from x^4 - 5x^3 -30 and you get -5x^3+3x^2-30 (a)

x^2 goes into -5x^3 -5x times

-5x(x^2-3) = -5x^3+15x

subtract this from (a) and you get 3x^2-15x-30 (b)

x^2 goes into 3x^2 3 times

3(x^2-3) = 3x^2-9

Subtract this from (b) and you get -15x - 21.

Therefore x^4 + 5x^3 - 30 / x^2 - 3 = x^2-5x+3 with remainder -15x-21.

Oops, I just realized thatyou had +5x and I used -5x. Anyway, use the same method and you'll get the correct answer.

Answer 4

x^2+5x+3
x^2-3\x^4 + 5x^3 - 30
x^4-3x^2
5x^3+3x^2
5x^3-15x
3x^2+15x
3x^2-9
15x-9

quotient is x^2+5x+3 remainder is 15x-9

Answer 5

x^2+3 remainder -21

Answer 6

.........[x^2+5x-3 + (-15x-39)/(x^2-3)]
x^2-3/x^4 + 5x^3 - 30
.........(-)x^4........(+)-3x^2
..................5x^3+3x^2
..................5x^3.........-15x
.........................-3x^2+15x-30
.........................-3x^2........+9
..................................(-15x -39)