What are the quotient and remainder of (x^5-x^3+-5) divided by (x-2)

Mathematics

2 answers:

pishuonlain [190]3 years ago

7 0

Answer:

x⁴+2x³ +3x²+6x +12 r. $\dfrac{19}{x-2}$

Step-by-step explanation:

You can do this through synthetic division. It is a shorthand way of dividing polynomials.

First you need to make your divisor equal to zero so you can solve for what goes into the division box:

x - 2 = 0 → x = 2

The next step is to arrange your polynomials in descending powers. All missing terms, you will put in a zero.

x⁵ - x³ + (-5) → x⁵ + 0x⁴- x³ + 0x² + 0x + (-5)

Now you can proceed to synthetic division. Make an upside down division box with the divisor outside and the coefficients of the dividend listed, along with their sign. Leave a space below the divident

+2 | +1 0 -1 0 0 -5

|

Next you bring down the first coefficient:

+2 | +1 0 -1 0 0 -5

|

Then you multiply it by the divisor and write the product under the next coefficient:

+2 | +1 0 -1 0 0 -5

| +2

Next add the column and put the sum below it:

+2 | +1 0 -1 0 0 -5

| +2

+1 +2

Then multiply again and repeat until you reach the last coefficient:

+2 | +1 0 -1 0 0 -5

| +2 +4 +6 +12 +24

+1 +2 +3 +6 +12 +19

Now that you have your results, add in the x and their powers. The powers will start with the highest power but 1 less than the dividend. Since the dividend's highest power is 5, then the quotient's highest power will be 4. Then write it in descending order :

+1x⁴ +2x³ +3x² +6x +12 +19

Now the last coefficient is your remainder. So your results will be:

x⁴+2x³ +3x²+6x +12 r. $\dfrac{19}{x-2}$