document.write( "Question 1200512: Consider the polynomial F(x)=4x^3+(a+1)x^2+x-5b and G(x)=4x^3-ax^2+bx-2, where a and b are constants. When both F(x) and G(x) are divided by x-1, the remainders are 3 and -3 respectively.
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Algebra.Com's Answer #834676 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "Consider polynomials F(x) = 4x^3+(a+1)x^2+x-5b and G(x) = 4x^3-ax^2+bx-2,
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\n" ); document.write( "When both F(x) and G(x) are divided by x-1, the remainders are 3 and -3, respectively.
\n" ); document.write( "a) Find the values of a and b.
\n" ); document.write( "b) Solve the equation F(x)-G(x) = -2.
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document.write( "(a)  We are given that F(x) = 4x^3+(a+1)x^2+x-5b gives the remainder 3 when divided by (x-1).\r\n" );
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document.write( "     According to the Remainder theorem, it means that  F(1) = 3.\r\n" );
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document.write( "     Calculate F(1) by substituting x= 1 into the formula for F(x)\r\n" );
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document.write( "        F(1) = 4*1 + (a+1)*1 + 1 - 5b = 4 + a+1 + 1 - 5b = a - 5b + 6.\r\n" );
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document.write( "     Hence, we have THIS equation for \"a\" and \"b\"\r\n" );
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document.write( "        a - 5b + 6 = 3,   or  a - 5b = -3.    (1)\r\n" );
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document.write( "(b)  Next, we are given that G(x) = 4x^3-ax^2+bx-2 gives the remainder -3 when divided by (x-1).\r\n" );
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document.write( "     According to the Remainder theorem, it means that  G(1) = -3.\r\n" );
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document.write( "     Calculate G(1) by substituting x= 1 into the formula for G(x)\r\n" );
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document.write( "        G(1) = 4*1 - a*1 + b*1 - 2 = 4 - a + b - 2 = -a + b + 2.\r\n" );
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document.write( "     Hence, we have THIS equation for \"a\" and \"b\"\r\n" );
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document.write( "        -a + b + 2 = -3,   or  -a + b = -5.    (2)\r\n" );
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document.write( "(c)  Thus we have two equations to find \"a\" and \"b\"\r\n" );
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document.write( "        a - 5b = -3    (1)\r\n" );
document.write( "       -a +  b = -5    (2)\r\n" );
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document.write( "     To solve, add the equations\r\n" );
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document.write( "           -4b = -8  ===>  b = (-8)/(-4) = 2.\r\n" );
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document.write( "     Then from equation (1),\r\n" );
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document.write( "        a = -3 + 5b = -3 + 5*2 = -3 + 10 = 7.\r\n" );
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document.write( "     +--------- ANSWER------------+\r\n" );
document.write( "     |     Thus  a = 7;  b = 2.   |\r\n" );
document.write( "     +----------------------------+\r\n" );
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\n" ); document.write( "\n" ); document.write( "First part is complete.\r
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\n" ); document.write( "\n" ); document.write( "The second part, after subtracting polynomials, gives a quadratic polynomial .\r
\n" ); document.write( "\n" ); document.write( "Working with it is simple arithmetic, so I leave this part on you.\r
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