document.write( "Question 1206675: list ALL roots (rational, irrational, and/or complex) of the given polynomial equation by using the methods discussed in this lesson. show work.\r
\n" ); document.write( "\n" ); document.write( "4x^3 + 16x^2 - 22x - 10 = 0 \n" ); document.write( "

Algebra.Com's Answer #844243 by greenestamps(13216) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( " $\"4x%5E3%2B16x%5E2-22x-10=0\"$
\n" ); document.write( " $\"2x%5E3%2B8x%5E2-11x-5=0\"$

\n" ); document.write( "The possible rational roots are 1, -1, 5, -5, 1/2, -1/2, 5/2, and -5/2.

\n" ); document.write( "You could test each possible root using synthetic division, but that would be a long slow process. Checking for zeros (roots) by evaluating the polynomial for each root is faster.

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document.write( "x=1:   2+8-11-5 = -6 --> not a root\r\n" );
document.write( "x=-1: -2+8+11-5 = 12 --> not a root\r\n" );
document.write( "x=5:   250+200-55-5 = 490 --> not a root\r\n" );
document.write( "x=-5: -250+200+55-5 = 0 --> -5 is a root

\n" ); document.write( "Divide the polynomial by (x-(-5)) to find the remaining quadratic polynomial.

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document.write( " -5 | 2   8 -11  -5\r\n" );
document.write( "    |   -10  10   5  \r\n" );
document.write( "    +---------------\r\n" );
document.write( "      2  -2  -1   0

\n" ); document.write( "This tells us that

\n" ); document.write( " $\"2x%5E3%2B8x%5E2-11x-5=%28x%2B5%29%282x%5E2-2x-1%29\"$

\n" ); document.write( "The quadratic factor is not factorable over the integers; find the other two roots using the quadratic formula. The other two (irrational) roots are

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\n" ); document.write( "ANSWERS: $\"-5\"$ , $\"%281%2Bsqrt%283%29%29%2F2\"$ , $\"%281-sqrt%283%29%29%2F2\"$

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