document.write( "Question 1049745: Hello,I have a question on polynomials..I've spent much time trying to figure out what to do,but I keep getting answers that don't add up in the end..Here is the question:\r
\n" ); document.write( "\n" ); document.write( "The polynomial f(x) is given by f(x) =4x^3 +ax^2 + bx+2. (4x+1) is a factor of f(x).When f(x) is divided by (x-1) there is a remainder of 20. Find a and b.\r
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Algebra.Com's Answer #665318 by advanced_Learner(501)\"\" \"About 
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Hello,I have a question on polynomials..I've spent much time trying to figure out what to do,but I keep getting answers that don't add up in the end..Here is the question:
\n" ); document.write( "The polynomial f(x) is given by f(x) =4x^3 +ax^2 + bx+2. (4x+1) is a factor of f(x).When f(x) is divided by (x-1) there is a remainder of 20. Find a and b.
\n" ); document.write( "Please help,I'm so stuck
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\n" ); document.write( "\n" ); document.write( "The polynomial f(x) is given by f(x) =\"4x%5E3+%2Bax%5E2+%2B+bx%2B2\". \"%284x%2B1%29\" is a factor of f(x).When f(x) is divided by \"%28x-1%29\" there is a remainder of \"20\". Find \"a\" and \"b\". \r
\n" ); document.write( "\n" ); document.write( "solution
\n" ); document.write( "first step
\n" ); document.write( "when f(x) is evaluated at x=\"-1%2F4\" the remainder is 0 Because \"4x%2B1\" is a factor of f(x).
\n" ); document.write( "f(x) =\"4x%5E3+%2Bax%5E2+%2B+bx%2B2\"
\n" ); document.write( "x=\"-1%2F4\"
\n" ); document.write( "\"0\" =\"4%2A%28-1%2F4%29%5E3+%2Ba%2A%28-1%2F4%29%5E2+%2B+b%2A%28-1%2F4%29%2B2\"\r
\n" ); document.write( "\n" ); document.write( "simplify to get
\n" ); document.write( "\"a-4b\"=\"-31\"\r
\n" ); document.write( "\n" ); document.write( "second step
\n" ); document.write( "what happens when f(x) is evaluated at x=\"1\"
\n" ); document.write( "f(x) =\"4x%5E3+%2Bax%5E2+%2B+bx%2B2\"\r
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\n" ); document.write( "\n" ); document.write( "\"4%2B+a+%2B+b%2B2\" this expression equals the remainder which we know to be \"20\"\r
\n" ); document.write( "\n" ); document.write( "\"20\"=\"4%2B+a+%2B+b%2B2\"
\n" ); document.write( "\"20\"=\"6%2B+a+%2B+b\"\r
\n" ); document.write( "\n" ); document.write( "\"20-6\"=\"+a+%2B+b\"\r
\n" ); document.write( "\n" ); document.write( "\"14\"=\"+a+%2B+b\"
\n" ); document.write( "\"a%2Bb\"=\"+14\"\r
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\n" ); document.write( "\n" ); document.write( "\"a-4b\"=\"-31\"
\n" ); document.write( "\"a%2Bb\"=\"+14\"\r
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Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
\n" ); document.write( " \"+system%28+%0D%0A++++1%5Ca+%2B+1%5Cb+=+14%2C%0D%0A++++1%5Ca+%2B+-4%5Cb+=+-31+%29%0D%0A++\"We'll use substitution. After moving 1*b to the right, we get:
\n" ); document.write( " \"1%2Aa+=+14+-+1%2Ab\", or \"a+=+14%2F1+-+1%2Ab%2F1\". Substitute that
\n" ); document.write( " into another equation:
\n" ); document.write( " \"1%2A%2814%2F1+-+1%2Ab%2F1%29+%2B+-4%5Cb+=+-31\" and simplify: So, we know that b=9. Since \"a+=+14%2F1+-+1%2Ab%2F1\", a=5.
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\n" ); document.write( " Answer: \"system%28+a=5%2C+b=9+%29\".
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\n" ); document.write( "\n" ); document.write( "answer
\n" ); document.write( "\"a\"=\"5\"
\n" ); document.write( "\"b\"=\"9\"
\n" ); document.write( "the polynomial is\r
\n" ); document.write( "\n" ); document.write( "f(x) =\"4x%5E3+%2B5x%5E2+%2B+9x%2B2\"\r
\n" ); document.write( "\n" ); document.write( "check that\r
\n" ); document.write( "\n" ); document.write( "\"%284x%5E3+%2B5x%5E2+%2B+9x%2B2%29\"/\"%284x%2B1%29\" gives off a remainder of \"0\" meaning
\n" ); document.write( "\"%284x%2B1%29\" is a factor of f(x).\r
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\n" ); document.write( "\n" ); document.write( "when
\n" ); document.write( "\"%284x%5E3+%2B5x%5E2+%2B+9x%2B2%29\"/\"%28x-1%29\" the remainder is \"20\" which is the same as evaluating the expression \"%284x%5E3+%2B5x%5E2+%2B+9x%2B2%29\" at x=\"1\"
\n" ); document.write( "\"%284+%2B5+%2B+9%2B2%29\"
\n" ); document.write( "\"%2820%29\"
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