document.write( "Question 216460: Find all real and complex zeros for $\"Find+all+real+and+complex+zeros+for+\"$ h(x)=x^3-x^2-7x-15\r
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\n" ); document.write( "Q:1 \n" ); document.write( "

Algebra.Com's Answer #163452 by jsmallt9(3758) $\"\"$ $\"About$

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Find all real and complex zeros for $\"h%28x%29=x%5E3-x%5E2-7x-15\"$
\n" ); document.write( "P: 1,3,5,15
\n" ); document.write( "Q:1

\n" ); document.write( "Since the function as you given it has no rational roots, I'm going to guess that you have a typo and $\"h%28x%29=x%5E3-x%5E2-7x%2B15\"$ . If this is wrong, then stop reading (and you may need to repost your question).

\n" ); document.write( "With this h(x) we can find that -3 is a rational root. To illustrate I'll use synthetic division because it will also show how to factor h(x):
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document.write( "-3 |    1   -1   -7    15\r\n" );
document.write( "----        -3   12   -15\r\n" );
document.write( "       -------------------\r\n" );
document.write( "        1   -4    5    0\r\n" );
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\n" ); document.write( "The remainder, 0, tells us that h(-3) = 0. So -3 is a root of h(x). And if -3 is a root of h(x), then (x+3) is a factor of h(x). And the other factor is found in the numbers in front of the reaminder above, \"1 -4 5\", which translates into $\"x%5E2+-4x+%2B+5\"$ .

\n" ); document.write( "So $\"h%28x%29+=+%28x%2B3%29%28x%5E2+-4x+%2B+5%29\"$ . The other roots for h(x) will come from the roots of $\"x%5E2-4x%2B5\"$ . This is a quadratic. Since it will not factor we'll use the quadratic formula:
\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29\"$
\n" ); document.write( "For our expression, a = 1, b = -4 and c = 5. Substituting these into the formula we get:
\n" ); document.write( " $\"x+=+%28-%28-4%29+%2B-+sqrt%28%28-4%29%5E2+-+4%281%29%285%29%29%29%2F%282%281%29%29\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"x+=+%284+%2B-+sqrt%2816+-+4%281%29%285%29%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%284+%2B-+sqrt%2816+-+20%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%284+%2B-+sqrt%28-4%29%29%2F2\"$
\n" ); document.write( "With the negative in the square root, we will have complex roots.
\n" ); document.write( " $\"x+=+%284+%2B-+sqrt%28-1%2A4%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%284+%2B-+sqrt%28-1%29%2Asqrt%284%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%284+%2B-+i%2A2%29%2F2\"$
\n" ); document.write( " $\"x+=+%282%282+%2B-+i%29%29%2F2\"$
\n" ); document.write( "The 2's cancel leaving:
\n" ); document.write( " $\"x+=+2+%2B-+i\"$
\n" ); document.write( "So our three roots are: -3, 2+i and 2-i \n" ); document.write( "

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