Question
Divide each of the following polynomials by synthetic division method and also by linear division method. Write the quotient and the remainder : $(y^3 – 216) ÷ (y – 6)$
✓
Answer
Synthetic division:
$\left(y^3-216\right) \div(y-6)$
Dividend $=y^3-216$
$\therefore$ Index form $= y ^3+0 y ^3+0 y -216$
$\therefore$ Coefficient form of dividend $=(1,0,0,-216)$
Divisor $=y-6$
$\therefore$ Opposite of $-6$ is $6$ .
Coefficient form of quotient $=(1,6,36)$
$\therefore$ Quotient $= y ^2+6 y +36$,
Remainder $=0$
Linear division method:
$y^3-216$
To get the term $y^3$, multiply $(y-6)$ by $y^2$ and add $6 y^2$,
$=y^2(y-6)+6 y^2-216$
$=y^2(y-6)+6 y \text { sup }>2-216$
To get the, term $6 y ^2$ multiply $(y-6)$ by $6 y$ and add 36 y ,
$=y^2(y-6)+6 y(y-6)+36 y-216$
$=y^2(y-6)+6 y(y-6)+36 y-216$
To get the term 36 y , multiply $( y -6$ ) by 36 and add 216 ,
$=y^2(y-6)+6 y(y-6)+36(y-6)+216-216$
$=(y-6)\left(y^2+6 y+36\right)+0$
Quotient $= y ^2+6 y +36$
Remainder $=0$
$\left(y^3-216\right) \div(y-6)$
Dividend $=y^3-216$
$\therefore$ Index form $= y ^3+0 y ^3+0 y -216$
$\therefore$ Coefficient form of dividend $=(1,0,0,-216)$
Divisor $=y-6$
$\therefore$ Opposite of $-6$ is $6$ .
Coefficient form of quotient $=(1,6,36)$
$\therefore$ Quotient $= y ^2+6 y +36$,
Remainder $=0$
Linear division method:
$y^3-216$
To get the term $y^3$, multiply $(y-6)$ by $y^2$ and add $6 y^2$,
$=y^2(y-6)+6 y^2-216$
$=y^2(y-6)+6 y \text { sup }>2-216$
To get the, term $6 y ^2$ multiply $(y-6)$ by $6 y$ and add 36 y ,
$=y^2(y-6)+6 y(y-6)+36 y-216$
$=y^2(y-6)+6 y(y-6)+36 y-216$
To get the term 36 y , multiply $( y -6$ ) by 36 and add 216 ,
$=y^2(y-6)+6 y(y-6)+36(y-6)+216-216$
$=(y-6)\left(y^2+6 y+36\right)+0$
Quotient $= y ^2+6 y +36$
Remainder $=0$
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