CPM Homework Help : PC Problem 7-62

Complete the following division problem. Express any remainder as a fraction.

$\frac { x ^ { 5 } - 1 } { x - 1 }$

Math Notes:

Choose one of the methods in the box below.

Using Long division:

$\require{enclose} \begin{array}{rll} x^3-4x^2-8x+2\ \\[-3pt] x-2 \enclose{longdiv}{x^4-6x^3+0x^2+18x-1}\kern-.2ex \\[-3pt] \underline{x^4-2x^3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \phantom{00}} && \\[-3pt] -4x^3+0x^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \phantom{0} \\[-3pt] \underline{\phantom{0}-4x^3+8x^2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \phantom{0}} \\[-3pt] \phantom{0}-8x^2+18x\ \ \ \ \ \ \ \ \\[-3pt] \underline{\phantom{0}-8x^2+16x \ \ \ \ \ \ \phantom{0}} \\[-3pt] {\phantom{0}2x-1 \ \ \phantom{0}} \\[-3pt] \underline{\phantom{0}2x-4 \ \ \phantom{0}} \\[-3pt] {\phantom{0}3 \ \ \phantom{0}} \\[-3pt] \text{ remainder } \uparrow \ \ \ \end{array}$

Final answer: $x^3-4x^2-8x+2+\frac{3}{x-2}$

Using Generic Rectangles:

2 by 5 generic rectangle

top edge first $x^3$

top edge second $-4x^2$

top edge third $-8x$

top edge fourth $+2$

top edge fifthRemainder

left edge top $x$

interior top first $x^4$

interior top second $-4x^3$

interior top third $-8x^2$

interior top fourth $+2x$

interior top fifth $3$

left edge bottom $-2$

interior bottom first $-2x^3$

interior bottom second $+8x^2$

interior bottom third $+16x$

interior bottom fourth $-4$

interior bottom fifth blank

$x^4$ Below generic rectangle

$-6x^3$

$+0x^2$

$+18x$

$-1$

Answer: $x^3-4x^2-8x+2+\frac{3}{x-2}$