When-the-expression-x-ax-bx-c-is-divided-by-x-4-the-remainder-is-18-x-and-when-it-is-divided-by-X-3-the-remainder-is-21-What-is-the-remainder-when-the-expression-is-divided-by-X-1-

Question Number 135928 by benjo_mathlover last updated on 17/Mar/21

$$$$When the expression x³+ax²+bx+c is divided by x²-4, the remainder is 18-x, and when it is divided by X+3 the remainder is 21. What is the remainder when the expression is divided by X+1?

Answered by john_santu last updated on 17/Mar/21

$$byHornermethod$$$$step\left(1\right)$$$$x^2=0x+4$$$$\overline{)\begin{array}{ccccc}\bullet & 1& a& b& c\\ 0& \ast & 0& 4& \\ 4& \ast & \ast & 0& 4a\\ \ast & 1& a& b+4& c+4a\end{array}}$$$$wegetremainder(b+4)x+c+4a\equiv -x+18$$$$so\{\begin{array}{l}b+4=-1\Rightarrow b=-5\\ c+4a=18\end{array}$$$$step\left(2\right)\Rightarrow p(-3)=-27+9a-5(-3)+c=21$$$$\to 9a+c=33.Solvingforaandc$$$$weget\{\begin{array}{l}a=3\\ c=6\end{array}\Rightarrow p\left(x\right)=x^3+3x^2-5x+6$$$$sotheremainderofp\left(x\right)when$$$$dividedbyx+1isp(-1)=-1+3+5+6=13$$

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