SOLUTION: What is the quotient? What is the remainder? (6x^2+31x-29)divided by (x+6)

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Question 1204167: What is the quotient?
What is the remainder?
(6x^2+31x-29)divided by (x+6)
Found 3 solutions by ikleyn, MathLover1, math_tutor2020:
Answer by ikleyn(52832)   (Show Source): You can put this solution on YOUR website!
.

According to the Remainder theorem, the remainder is the value of the polynomial at x = -6


    remainder = 6*(-6)^2 + 31*(-6) - 29 = 1.



Next, make these identical transformations 

    6x^2 + 31x - 29 = (6x^2 + 36x) - (5x + 29) = 6x*(x+6) - (5x+30) + 1 = 6x*(x+6) - 5(x+6) + 1 = (6x-5)*(x+6) + 1.


It shows that the quotient is 6x-5 and the remainder is 1.    ANSWER

Solved.



Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!


use long division
.........(=>quotient
|
....................subtract
.........................bring down
..................
..........................subtract
........................... =>reminder

answer:
the quotient is
the reminder is


Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answers:
Quotient = 6x-5
Remainder = 1

Explanation

The other tutors have great approaches.
I'll use synthetic division as another alternative.

The numerator polynomial is 6x^2+31x-29.
The coefficients are placed along the top row of the synthetic division table.

To the left of those coefficients is the test root x = -6, which is derived from solving x+6 = 0.

We have this so far
-6631-29

Pull down the leading coefficient 6 to place it in the bottom row.
-6631-29
6

Multiply that value (6) by the test root (-6).
The result -36 is placed just under the 31.
-6631-29
-36
6

Then we add 31 to -36 to get -5. That result is placed under the -36.
-6631-29
-36
6-5

The previous two steps are repeated to fill out the last column as shown below
-6631-29
-3630
6-51

The last value in the bottom row is the remainder.
The remainder is 1

The other values in the bottom row are the coefficients of the quotient.
The quotient is 6x-5

This will mean
(6x^2+31x-29)/(x+6) = 6x-5 remainder 1

We can rewrite that as


Then we can multiply both sides by the LCD (x+6) to get


These claims can be verified using a tool like WolframAlpha or the CAS feature in GeoGebra.
Many other calculators online offer similar capabilities.

Another way to verify is to expand out the right hand side of the last equation we mentioned.
6x^2+31x-29 = (x+6)(6x-5) + 1
6x^2+31x-29 = x(6x-5)+6(6x-5) + 1
6x^2+31x-29 = (6x^2-5x)+(36x-30) + 1
6x^2+31x-29 = 6x^2-5x+36x-30 + 1
6x^2+31x-29 = 6x^2+31x-29
The answer is confirmed.
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