document.write( "Question 804181: What are the real zeros of f(x)=x^3-x^2-37x-35 \n" ); document.write( "

Algebra.Com's Answer #484649 by solver91311(24713) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Properties of Polynomials:\r
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\n" ); document.write( "\n" ); document.write( "The Fundamental Theorem of Algebra guarantees that an

-th degree polynomial function will have

zeros, counting multiplicities, some or all of which may be complex roots.\r
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\n" ); document.write( "\n" ); document.write( "Complex roots always appear in conjugate pairs, so:\r
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\n" ); document.write( "\n" ); document.write( "If

is even, the polynomial may have 0, 2, 4, ... , n real number zeros.\r
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\n" ); document.write( "\n" ); document.write( "If

is odd, the polynomial is guaranteed to have at least one real number zero, but may have 1, 3, 5, ..., n real number zeros.\r
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\n" ); document.write( "\n" ); document.write( "Rational Roots Theorem: If a polynomial has a rational number zero, then it will be of the form

where

is an integer factor of the constant cooefficient,

, and

is an integer factor of the lead coefficient,

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\n" ); document.write( "\n" ); document.write( "Corollary to The Fundamental Theorem of Algebra: If

is a zero of a polynomial, then

is a factor of the polynomial.\r
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\n" ); document.write( "\n" ); document.write( "is a third degree polynomial function, hence there is either 1 real zero or there are 3 real zeros. According to the Rational Roots Theorem, if any of these zeros are rational numbers, they are

, or

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\n" ); document.write( "\n" ); document.write( "We test using Synthetic Division. First test 1:
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document.write( "1  |  1   -1   -37   -35\r\n" );
document.write( "   |       1     0   -37\r\n" );
document.write( "   ----------------------\r\n" );
document.write( "      1    0   -37   -72  Last result non-zero, 1 is not a zero.\r\n" );
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document.write( "Try -1\r\n" );
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document.write( "-1 |  1   -1   -37   -35\r\n" );
document.write( "   |      -1     2    35\r\n" );
document.write( "   ----------------------\r\n" );
document.write( "      1   -2   -35     0  Last result is zero, -1 is a zero\r\n" );
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\n" ); document.write( "Hence

is a factor and

is the other factor. But

is a factorable quadratic.\r
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\n" ); document.write( "\n" ); document.write( "I'll leave it to you to finish this.\r
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\n" ); document.write( "\n" ); document.write( "John
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\n" ); document.write( "Egw to Beta kai to Sigma
\n" ); document.write( "My calculator said it, I believe it, that settles it
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$\"The$

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