Let f(x)=4x4+12x3−9x2+x+3 and d(x)=x2+3x−2. If f(x)=q(x)d(x)+r(x) for some polynomials q(x) and r(x) where degr<degd, calculate q(1)+r(−1).
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\multicolumn2r4x2\cline2−6x2+3x−2\multicolumn2r−4x4\cline2−4\multicolumn2r0\multicolumn2r\cline4−6\multicolumn2r−14x4−12x30+12x3+8x2−x2+x20−9x2+x+3x4x+x+3−2+1+3Since degd>deg(4x+1) we cannot divide any further. So, q(x)=4x2−1 and r(x)=4x+1. Then q(1)+r(−1)=4(1)2+1+4(−1)−1=0.