document.write( "Question 156591: 1.x^4=5x^2
\n" ); document.write( "2.x^4+18x^3+8x+144=0
\n" ); document.write( "3.x^3+19x^2-7x-133=0
\n" ); document.write( "4.x^6-13x^5-7x=-91
\n" ); document.write( "I need help solving these. So far I have spent hours on just these problems our teacher doesn't explain things very well.
\n" ); document.write( "Thanks
\n" ); document.write( "Lauren \n" ); document.write( "
Algebra.Com's Answer #115386 by Earlsdon(6294)\"\" \"About 
You can put this solution on YOUR website!
Solve for x:
\n" ); document.write( "1) \"x%5E4+=+5x%5E2\" Divide both sides by \"x%5E2\"
\n" ); document.write( "\"x%5E4-5x%5E2+=+0\" Factor out an \"x%5E2\"
\n" ); document.write( "\"x%5E2%28x%5E2-5%29+=+0\" Apply the zero product rule: If a*b = 0, then either a = 0 or b = 0 or both.
\n" ); document.write( "\"x%5E2+=+0\" or \"x%5E2-5+=+0\"
\n" ); document.write( "If \"x%5E2+=+0\" then \"x+=+0\" and \"x+=+0\"
\n" ); document.write( "If \"x%5E2-5+=+0\" then \"x%5E2+=+5\" Take the square root of both sides to get:
\n" ); document.write( "\"x+=+sqrt%285%29\" or \"x+=+-sqrt%285%29\"
\n" ); document.write( "Now because this is a fourth degree polynomial, you can expect four solutions, some of which may be identical.
\n" ); document.write( "The solutions are:
\n" ); document.write( "\"x+=+0\"
\n" ); document.write( "\"x+=+0\"
\n" ); document.write( "\"x+=+sqrt%285%29\"
\n" ); document.write( "\"x+=+-sqrt%285%29\"
\n" ); document.write( "--------------------------------
\n" ); document.write( "2) \"x%5E4%2B18x%5E3%2B8x%2B144+=+0\" This polynomial can be factored into:
\n" ); document.write( "\"%28x%5E3%2B8%29%28x%2B18%29+=+0\" and the first factor can be further factored as it is the sum of two cubes \"%28x%29%5E3\" and \"2%5E3\", so we get:
\n" ); document.write( "\"%28x%2B2%29%28x%5E2-2x%2B4%29%28x%2B18%29+=+0\" Now, applying the zero product rule that was used in the first problem, you will set each of these three factors equal to zero.
\n" ); document.write( "\"x%2B2+=+0\" Here, \"x+=+-2\"
\n" ); document.write( "\"x%5E2-2x%2B4+=+0\" Here, we'll have to use the quadratic formula to solve for x: \"x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a\" and a = 1, b = -2, and c = 4, so...
\n" ); document.write( "\"x+=+%28-%28-2%29%2B-sqrt%28%28-2%29%5E2-4%281%29%284%29%29%29%2F2%281%29\" Simplifying this, we get:
\n" ); document.write( "\"x+=+%282%2B-sqrt%284-16%29%29%2F2\"
\n" ); document.write( "\"x+=+%282%2B-sqrt%28-12%29%29%2F2\"
\n" ); document.write( "\"x+=+%282%2B-sqrt%284%28-3%29%29%29%2F2\"
\n" ); document.write( "\"x+=+%282%2B2%2Asqrt%28-3%29%29%2F2\" or \"x+=+%282-2%2Asqrt%28-3%29%29%2F2\" Simplifying this:
\n" ); document.write( "\"x+=+1%2Bsqrt%283%29i\" or \"x+=+1-sqrt%283%29i\"
\n" ); document.write( "...and for the last one:
\n" ); document.write( "\"x+%2B+18+=+0\"
\n" ); document.write( "\"x+=+-18\"
\n" ); document.write( "So, as in the first problem, because you have a fourth degree polynomial, you can expect four solutions, and they are:
\n" ); document.write( "\"x+=+-2\"
\n" ); document.write( "\"x+=+1%2Bsqrt%283%29i\"
\n" ); document.write( "\"x+=+1-sqrt%283%29i\"
\n" ); document.write( "\"x+=+-18\"
\n" ); document.write( "-----------------
\n" ); document.write( "3) \"x%5E3%2B19x%5E2-7x-133+=+0\" You can solve this by \"factoring by grouping\"
\n" ); document.write( "\"%28x%5E3%2B19x%29-%287x%2B133%29+=+0\" Now factor each of the two terms:
\n" ); document.write( "\"x%5E2%28x%2B19%29-7%28x%2B19%29+=+0\" Factor out the common factor of \"x%2B19\"
\n" ); document.write( "\"%28x%2B19%29%28x%5E2-7%29+=+0\" Apply the zero product rule.
\n" ); document.write( "\"x%2B19+=+0\" or \"x%5E2-7+=+0\" so...
\n" ); document.write( "\"x+=+-19\" or \"x%5E2+=+7\"
\n" ); document.write( "\"x%5E2+=+7\" Take the square root of both sides.
\n" ); document.write( "\"x+=+sqrt%287%29\" or \"x+=+-sqrt%287%29\"
\n" ); document.write( "The three solutions are:
\n" ); document.write( "\"x+=+-19\"
\n" ); document.write( "\"x+=+sqrt%287%29\"
\n" ); document.write( "\"x+=+-sqrt%287%29\"
\n" ); document.write( "----------------
\n" ); document.write( "I'll get you started on the last one but see if you can finish it up yourself.
\n" ); document.write( "4) \"x%5E6-13x%5E5-7x+=+-91\" Add 91 to both sides.
\n" ); document.write( "\"x%5E6-13x%5E5-7x-91+=+0\" Factor by grouping.
\n" ); document.write( "\"%28x%5E6-13x%5E5%29-%287x-91%29+=+0\" Factor an \"x%5E5\" from the first term and a 7 from the second term.
\n" ); document.write( "\"x%5E5%28x-13%29-7%28x-13%29+=+0\" Now factor the common factor \"%28x-13%29\"
\n" ); document.write( "\"%28x-13%29%28x%5E5-7%29+=+0\" Apply the zero product rule.
\n" ); document.write( "\"x-13+=+0\" or \"x%5E5-7+=+0\"
\n" ); document.write( "Can you finish this? \n" ); document.write( "
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