document.write( "Question 1167586: given that x=3+i is a root of p(x)=3x^4+ax^3+34x^2+bx-20 find the value of a and b if when p(-1)=34. then factor the polynomial completely \n" ); document.write( "

Algebra.Com's Answer #792182 by greenestamps(13198) $\"\"$ $\"About$

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\n" ); document.write( "The information p(-1)=34 is not needed to solve the problem....

\n" ); document.write( "Since x=3+i is a root, x=3-i is another root. The quadratic factor corresponding to those two roots is $\"x%5E2-6x%2B10\"$

\n" ); document.write( "Given the leading coefficient and constant term of the polynomial, we know the factorization of the polynomial is of the form

\n" ); document.write( " $\"%28x%5E2-6x%2B10%29%283x%5E2%2Bnx-2%29\"$

\n" ); document.write( "Performing that multiplication yields

\n" ); document.write( " $\"3x%5E4%2B%28n-18%29x%5E3%2B%2828-6n%29x%5E2%2B%2810n%2B12%29x-20\"$

\n" ); document.write( "We can solve for n knowing that the coefficient of the quadratic term is 34

\n" ); document.write( " $\"28-6n+=+34\"$
\n" ); document.write( " $\"6n+=+-6\"$
\n" ); document.write( " $\"n+=+-1\"$

\n" ); document.write( "We can then find the values of a and b, which are the coefficients of the cubic and linear terms of the polynomial:

\n" ); document.write( "a = n-18 = -19
\n" ); document.write( "b = 10n+12 = 2

\n" ); document.write( "ANSWERS: a = -19; b = 2
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