Question
Divide each of the following polynomials by synthetic division method and also by linear division method. Write the quotient and the remainder : $(x^4 – 3x^2 – 8) ÷ (x + 4)$
✓
Answer
$\left(x^4-3 x^2-8\right)+(x+4)$
Dividend $=x^4-3 x^2-8$
$\therefore$ Index form $= x ^4+0 x ^3-3 x ^2+0 x -8$
$\therefore$ Coefficient form of the dividend $=(1,0,-3,0,-8)$
Divisor $=x+4$
$\therefore $ Opposite of $+ 4$ is $-4$
$\therefore$ Coefficient form of quotient $=(1,-4,13,-52)$
$\therefore$ Quotient $= x ^3-4 x ^2+13 x -52$,
Remainder $=200$
Linear division method:
$x^4-3 x^2-8$
To get the term $x^4$, multiply $(x+4)$ by $x^3$ and subtract $4 x^3$,
$=x^3(x+4)-4 x^3-3 x^2-8$
$=x^3(x+4)-4 x^3-3 x^2-8$
To get the term $-4 x^3$, multiply $(x+4)$ by $-4 x^2$ and add $16 x^2$,
$=x^3(x+4)-4 x^2(x+4)+16 x^2-3 x^2-8$
$=x^3(x+4)-4 x^2(x+4)+13 x^2-8$
To get the term $13 x^2$, multiply $(x+4)$ by $13 x$ and subtract $52 x$,
$=x^3(x+4)-4 x^2(x+4)+13 x(x+4)-52 x-8$
$=x^3(x+4)-4 x^2(x+4)+13 x(x+4)-52 x-8$
To get the term $-52 x$, multiply $(x+4)$ by -52 and add 208 ,
$=x^3(x+4)-4 x^2(x+4)+13 x(x+4)-52(x+4)+208-8$
$=(x+4)\left(x^3-4 x^2+13 x-52\right)+200$
$\therefore \text { Quotient }=x^3-4 x^2+13 x-52$
Remainder $200$
Dividend $=x^4-3 x^2-8$
$\therefore$ Index form $= x ^4+0 x ^3-3 x ^2+0 x -8$
$\therefore$ Coefficient form of the dividend $=(1,0,-3,0,-8)$
Divisor $=x+4$
$\therefore $ Opposite of $+ 4$ is $-4$
$\therefore$ Coefficient form of quotient $=(1,-4,13,-52)$
$\therefore$ Quotient $= x ^3-4 x ^2+13 x -52$,
Remainder $=200$
Linear division method:
$x^4-3 x^2-8$
To get the term $x^4$, multiply $(x+4)$ by $x^3$ and subtract $4 x^3$,
$=x^3(x+4)-4 x^3-3 x^2-8$
$=x^3(x+4)-4 x^3-3 x^2-8$
To get the term $-4 x^3$, multiply $(x+4)$ by $-4 x^2$ and add $16 x^2$,
$=x^3(x+4)-4 x^2(x+4)+16 x^2-3 x^2-8$
$=x^3(x+4)-4 x^2(x+4)+13 x^2-8$
To get the term $13 x^2$, multiply $(x+4)$ by $13 x$ and subtract $52 x$,
$=x^3(x+4)-4 x^2(x+4)+13 x(x+4)-52 x-8$
$=x^3(x+4)-4 x^2(x+4)+13 x(x+4)-52 x-8$
To get the term $-52 x$, multiply $(x+4)$ by -52 and add 208 ,
$=x^3(x+4)-4 x^2(x+4)+13 x(x+4)-52(x+4)+208-8$
$=(x+4)\left(x^3-4 x^2+13 x-52\right)+200$
$\therefore \text { Quotient }=x^3-4 x^2+13 x-52$
Remainder $200$
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