document.write( "Question 1204167: What is the quotient?
\n" ); document.write( "What is the remainder?\r
\n" ); document.write( "\n" ); document.write( "(6x^2+31x-29)divided by (x+6) \n" ); document.write( "

Algebra.Com's Answer #840239 by math_tutor2020(3817) $\"\"$ $\"About$

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\n" ); document.write( "Answers:
\n" ); document.write( "Quotient = 6x-5
\n" ); document.write( "Remainder = 1\r
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\n" ); document.write( "\n" ); document.write( "Explanation\r
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\n" ); document.write( "\n" ); document.write( "The other tutors have great approaches.
\n" ); document.write( "I'll use synthetic division as another alternative.\r
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\n" ); document.write( "\n" ); document.write( "The numerator polynomial is 6x^2+31x-29.
\n" ); document.write( "The coefficients are placed along the top row of the synthetic division table.\r
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\n" ); document.write( "\n" ); document.write( "To the left of those coefficients is the test root x = -6, which is derived from solving x+6 = 0.\r
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\n" ); document.write( "\n" ); document.write( "We have this so far
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-6	6	31	-29

\n" ); document.write( "Pull down the leading coefficient 6 to place it in the bottom row.
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-6	6	31	-29

	6

\n" ); document.write( "Multiply that value (6) by the test root (-6).
\n" ); document.write( "The result -36 is placed just under the 31.
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-6	6	31	-29
		-36
	6

\n" ); document.write( "Then we add 31 to -36 to get -5. That result is placed under the -36.
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-6	6	31	-29
		-36
	6	-5

\n" ); document.write( "The previous two steps are repeated to fill out the last column as shown below
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-6	6	31	-29
		-36	30
	6	-5	1

\n" ); document.write( "The last value in the bottom row is the remainder.
\n" ); document.write( "The remainder is 1\r
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\n" ); document.write( "\n" ); document.write( "The other values in the bottom row are the coefficients of the quotient.
\n" ); document.write( "The quotient is 6x-5\r
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\n" ); document.write( "\n" ); document.write( "This will mean
\n" ); document.write( "(6x^2+31x-29)/(x+6) = 6x-5 remainder 1\r
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\n" ); document.write( "\n" ); document.write( "We can rewrite that as
\n" ); document.write( " $\"%286x%5E2%2B31x-29%29%2F%28x%2B6%29+=+6x-5+%2B+1%2F%28x%2B6%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Then we can multiply both sides by the LCD (x+6) to get
\n" ); document.write( " $\"6x%5E2%2B31x-29+=+%28x%2B6%29%286x-5%29+%2B+1\"$ \r
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\n" ); document.write( "\n" ); document.write( "These claims can be verified using a tool like WolframAlpha or the CAS feature in GeoGebra.
\n" ); document.write( "Many other calculators online offer similar capabilities.\r
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\n" ); document.write( "\n" ); document.write( "Another way to verify is to expand out the right hand side of the last equation we mentioned.
\n" ); document.write( "6x^2+31x-29 = (x+6)(6x-5) + 1
\n" ); document.write( "6x^2+31x-29 = x(6x-5)+6(6x-5) + 1
\n" ); document.write( "6x^2+31x-29 = (6x^2-5x)+(36x-30) + 1
\n" ); document.write( "6x^2+31x-29 = 6x^2-5x+36x-30 + 1
\n" ); document.write( "6x^2+31x-29 = 6x^2+31x-29
\n" ); document.write( "The answer is confirmed.
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