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

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.

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 3x³First 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,

Lines that intersect at one point

122-62 because that is subtraction.