document.write( "Question 672928: Use division to show that the indicated polynomial is a factor of the given polynomial function f.
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Algebra.Com's Answer #418429 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!
(x - 1)(x - 4); f(x) = x⁴ - 5x³ + 13x² - 45x + 36
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document.write( "We first find  of \r\n" );
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document.write( "to show that the remainder is 0.  That will show that (x - 1) \r\n" );
document.write( "is a factor of f(x).  I'll use synthetic division where we use +1\r\n" );
document.write( "to divide by (x - 1):\r\n" );
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document.write( " 1 | 1 -5 13 -45  36 \r\n" );
document.write( "   |    1 -4   9 -36\r\n" );
document.write( "     1 -4  9 -36   0\r\n" );
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document.write( "So therefore (x - 1) is a factor, and we have now factored f(x) as\r\n" );
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document.write( "f(x) = (x - 1)(x³ - 4x² + 9x - 36)\r\n" );
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document.write( "Now we must also show that the factor in the second parentheses is\r\n" );
document.write( "divisible by (x - 1)\r\n" );
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document.write( "Now we must show that (x - 4) is also a fsctor\r\n" );
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document.write( "We now find  \r\n" );
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document.write( "to show that the remainder is 0.  That will show that (x - 4) \r\n" );
document.write( "is a factor of x³ - 4x² + 9x - 36, and therefore a factor of\r\n" );
document.write( "f(x).  I'll use synthetic division again this time where we \r\n" );
document.write( "use +4 to divide by (x - 4):\r\n" );
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document.write( " 4 | 1 -4  9 -36 \r\n" );
document.write( "   |    4  0  36\r\n" );
document.write( "     1  0  9   0\r\n" );
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document.write( "So therefore (x - 4) is a factor, and we have now factored f(x) as\r\n" );
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document.write( "We started with\"\r\n" );
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document.write( "f(x) = x⁴ - 5x³ + 13x² - 45x + 36\r\n" );
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document.write( "and factored it with synthetic division as this\r\n" );
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document.write( "f(x) = (x - 1)(x³ - 4x² + 9x - 36)\r\n" );
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document.write( "And further factored the right factor with synthetic\r\n" );
document.write( "division as this:\r\n" );
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document.write( "f(x) = (x - 1)(x - 4)(x² + 0x + 9)\r\n" );
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document.write( "Eliminating the zero term, this is the factored form\r\n" );
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document.write( "f(x) = (x - 1)(x - 4)(x² + 9)\r\n" );
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document.write( "which proves that (x - 1)(x - 4) is a factor of f(x).\r\n" );
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document.write( "Edwin

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