document.write( "Question 1183613: The expression ax^3 + bx^2 - 5x + 2a is exactly divisible by x^2 - 3x - 4. Calculate the value of a and of b and factorise the expression completely. \n" ); document.write( "

Algebra.Com's Answer #814018 by KMST(5328) $\"\"$ $\"About$

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$\"x%5E2-3x-4=%28x%2B1%29%28x-4%29\"$ tells us
\n" ); document.write( "1) two of the factors we need to \"factorise the expression completely\" and
\n" ); document.write( "2) that the expression is zero for $\"x=-1\"$ and for $\"x=4\"$ .\r
\n" ); document.write( "\n" ); document.write( "ONE OPTION:
\n" ); document.write( "Using the second finding we know that
\n" ); document.write( " $\"a%28-1%29%5E3%2Bb%28-1%29%5E2-5%28-1%29%2B2a=0\"$ --> $\"-a%2Bb%2B5%2B2a=0\"$ --> $\"a%2Bb%2B5=0\"$ and
\n" ); document.write( " $\"a%284%29%5E3%2Bb%284%29%5E2-5%284%29%2B2a=0\"$ --> $\"64a%2B16b-20%2B2a=0\"$ --> $\"66a%2B16b-20=0\"$
\n" ); document.write( "We can then solve the system of equations
\n" ); document.write( " $\"system%28a%2Bb%2B5=0%2C66a%2B16b-20=0%29\"$ or $\"system%28a%2Bb=-5%2C66a%2B16b=20%29\"$
\n" ); document.write( "to find
\n" ); document.write( " $\"system%28a=2%2Cb=-7%29\"$ ;
\n" ); document.write( "then re-write the original expression/polynomial as
\n" ); document.write( " $\"2x%5E3-7x%5E2-5x%2B4\"$ ,
\n" ); document.write( "and then find the other factor(s) we need to \"factorise the expression completely\" as
\n" ); document.write( " $\"%282x%2Bc%29%28x%2B1%29%28x-4%29\"$ or as $\"2%28x-r%29%28x%2B1%29%28x-4%29\"$ ,
\n" ); document.write( "depending on what is understood by factorise completely.
\n" ); document.write( "The missing factor(s) can be found by
\n" ); document.write( "1) dividing the original expression by $\"x%5E2-3x-4\"$ , or by
\n" ); document.write( "2) dividing in two steps by $\"%28x%2B1%29\"$ and then by $\"%28x-4%29\"$ , or by
\n" ); document.write( "3) multiplying $\"%282x%2Bc%29%28x%5E2-3x-4%29\"$ or as $\"2%28x-r%29%28x%5E2-3x-4%29\"$
\n" ); document.write( "and knowing that identical expressions means equal for every value of the variable, comparing to
\n" ); document.write( " $\"2x%5E3-7x%5E2-5x%2B4\"$
\n" ); document.write( "to conclude that the the two ways to express the independent term mean
\n" ); document.write( " $\"-4c=4\"$ --> $\"c=-1\"$ or $\"2r=4\"$ --> $\"r=1%2F2\"$
\n" ); document.write( "
\n" ); document.write( "ANOTHER OPTION:
\n" ); document.write( "Multiply $\"%28ax%2Bc%29%28x%5E2-3x-4%29\"$ or as $\"a%28x-r%29%28x%5E2-3x-4%29\"$
\n" ); document.write( "and match the expressions of the coefficients in that product to the coefficients in
\n" ); document.write( " $\"ax%5E3+%2B+bx%5E2+-+5x+%2B+2a\"$ to find $\"a\"$ , $\"b\"$ , and either $\"c\"$ or $\"r\"$
\n" ); document.write( "For example:
\n" ); document.write( " $\"%28ax%2Bc%29%28x%5E2-3x-4%29\"$ $\"%22=%22\"$ $\"ax%5E3-3ax%5E2-4ax%2Bcx%5E2-3cx-4c\"$
\n" ); document.write( " $\"%22=%22\"$ $\"ax%5E3%2B%28-3a%2Bc%29x%5E2%2B%28-4a-3c%29x-4c\"$ gives you
\n" ); document.write( " $\"system%28a=a%2Cb=-3a%2Bc%2C-5=-4a-3c%2C2a=-4c%29\"$ or $\"system%28-5=-4a-3c%2C2a=-4c%29\"$ and $\"b=-3a%2Bc\"$
\n" ); document.write( "which tells you
\n" ); document.write( " $\"system%28a=2%2Cc=-1%29\"$ and substituting into $\"b=-3a%2Bc\"$ you get $\"b=-7\"$ \n" ); document.write( "

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