Question

# The polynomial p(x)=x4−2x3+3x2−ax+b when divided by (x-1) and (x +1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when p(x) is divided by (x -2).

Solution

## Let:p(x)=x4−2x3+3x2−ax+b Now,When px is divided by x-1, the remainder is p1.When px is divided by x+1, the remainder is p-1. Thus, we have: p(1)=14−2×13+3×12−a×1+b=1−2+3−a+b=2−a+b And, p(−1)=−14−2×−13+3×−12−a×−1+b=1+2+3+a+b=6+a+b Now, 2−a+b=5−−−−(1) 6+a+b=19−−−(2)​​​​​​​ Adding (1) and (2), we get: 8+2b=24⇒2b=16⇒b=8 By putting the value of b, we get the value of a, i.e., 5. ∴ a = 5 and b = 8 Now, p(x)=x4−2x3+3x2−5x+8 Also, When px is divided by x-2, the remainder is p(2). Thus we have: p(2)=24−2×23+3×22−5×2+8=16−16+12−10+8=10 MathematicsSecondary School Mathematics IXStandard IX

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