Question

(9x4 + 3x3y - 5x2y2 + xy3) ÷ (3x3 + 2x2y - xy2)

Answer 1

Answer:

3x-y

Step-by-step explanation:

• This is a polynomial of 4th order.
• To obtain the quotient, we use long division method

First step; in using long division method we look for a number which will multiply the first term of the denominator (divisor) to obtain exactly the first term of the numerator (dividend).

• In this question, First term of the divisor is 3x³
• First term of the numerator is 9x⁴

Hence, 3x³ multiply 3x = 9x⁴

Second step; multiply all the divisor by 3x and subtract the result from the dividend.

3x(3x³ + 2x²y - xy²) = 9x⁴ + 6x³y - 3x²y²

Subtract 9x⁴ + 6x³y - 3x²y² from 9x⁴ + 3x³y - 5x²y² + xy³

= 9x⁴ + 3x³y - 5x²y² + xy³ - (9x⁴ + 6x³y - 3x²y²)

= 9x⁴ + 3x³y - 5x²y² + xy³ - 9x⁴ - 6x³y +3x²y²

= -3x³y - 2x²y² + xy³ (this is the remainder after subtraction and becomes the new dividend)

Third step; note, the first term of the dividend has reduced to -3x³y, again think of a number that will multiply the first term of the divisor to obtain exactly -3x³y.

The number or term is '-y';      -y(3x³) = -3x³y

Fourth step; multiply all the divisor by -y and subtract the result from the dividend.

-y (3x³ + 2x²y - xy²) = -3x³y - 2x²y² + xy³

Subtract -3x³y - 2x²y² + xy³ from -3x³y - 2x²y² + xy³

= -3x³y - 2x²y² + xy³ - (-3x³y - 2x²y² + xy³)

= -3x³y - 2x²y² + xy³ +3x³y + 2x²y²- xy³

= 0

The remainder is now zero (0), we stop dividing

Note: All the terms used in multiplying the divisor before subtracting it from the dividend is the quotient of the long division.

HENCE, $\frac{9x^{4} +3x^{3}y-5x^{2}y^{2} + xy^{3}}{3x^{3} + 2x^{2}y -xy^{2}} = 3x-y$

Answer 2

We are asked to perform long division of the given problem where the expression 9x4 + 3x3y - 5x2y2 + xy3 is to be divided by the expression 3x3 + 2x2y - xy2 

The solution is shown below:
                                 3x  + y  
                              --------------------------------------
3x3 + 2x2y -xy2 √ 9x4 + 3x3y - 5x2y2 + xy3
                            - ( 9x4 + 6x3y - 3x2y2)
                            -----------------------------------------
                                 0    -  3x3y  - 2x2y2  +  xy3
                                       + (3x3y + 2x2y2 - xy3 )
                              ---------------------------------------
                                          0
The answer is 3x + y. No remainder.