Fo the equation 5x^5 + 13x^4 - 2x^2 -6, x-1 , How to divide the first expression by the second?

Can someone please give me a shortcut or a explanation/methods I can apply for similar complicated question like this one (if possible). Thank you.

2 Answers
Jan 22, 2018

(5x^5+13x^4-2x^2-6)/(x-1) = 5x^4+18x^3+18x^2+16x+16+10/(x-1)

Explanation:

One method is to redistribute the terms in the dividend into multiples of the divisor, which you can do like this:

5x^5+13x^4-2x^2-6

=5x^5-5x^4+18x^4-2x^2-6

=5x^5-5x^4+18x^4-18x^3+18x^3-2x^2-6

=5x^5-5x^4+18x^4-18x^3+18x^3-18x^2+16x^2-6

=5x^5-5x^4+18x^4-18x^3+18x^3-18x^2+16x^2-16x+16x-6

=5x^5-5x^4+18x^4-18x^3+18x^3-18x^2+16x^2-16x+16x-16+10

=(x-1)(5x^4+18x^3+18x^2+16x+16)+10

So:

(5x^5+13x^4-2x^2-6)/(x-1) = 5x^4+18x^3+18x^2+16x+16+10/(x-1)

Alternative method

The way I would do the division would be to start writing the factorisation and add terms one at a time...

The first term of the quotient must be 5x^4 in order that when multiplied by x it gives the leading term 5x^5, so start to write down:

5x^5+13x^4-2x^2-6 = (x-1)(5x^4...

Then note that (-1)(5x^4) will give us -5x^4, but we want 13x^4. So we can deduce that the next term in the quotient is 18x^3:

5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3...

This will result in a term -18x^3 which we don't want, so the next term in our quotient is 18x^2...

5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3+18x^2...

This will result in a term -18x^2, but we want -2x^2. So the next term we need for our quotient is 16x...

5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3+18x^2+16x...

This will give result in -16x, but there is no x term in the dividend, so we need a constant term 16 to cancel it out...

5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3+18x^2+16x+16)...

Finally, note that (-1)(16) = -16, but we want -6, so we need to add a remainder term 10...

5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3+18x^2+16x+16)+10

This takes a lot longer to describe than to do. With a bit of practice you may find this the most convenient method.

Jan 23, 2018

Quotient 5x^4 +18x^3 +18x^2 +16x Remainder 10.

Explanation:

Shortcut method is synthetic division. To do this, write the coefficients of all the terms, including the missing ones, starting with the highest degree term in a single row. The missing terms here are x^3 and x. Their coefficients would be o s. To the left of this row write the value of x ,obtained by equating the divisor to 0. In this case it would be x=1

            1|   5         13           0         -2       0           -6

 Now divide the first number 5 by 1 and the write the quotient 5 below the next number 13.

Add them to get 18.

Now divide this number by 1 and write the quotient 18 below the next number 0. continue this process till the last digit. The final display would be as follows

            1|   5         13           0          -2      0            -6
                            _5______18____18___16_____16___________
                             18           18        16     16          10

The Last digit obtained is 10. This is the Remainder. The quotient would be 5x^4 +18x^3 +18x^2 +16x +16