Practice 2 Solutions

Simplify. Write the polynomial expression with positive exponents. Write the remainder as a fraction if one exists.

$(x^{4} - 8 x^{3} + 15 x^{2} - 23 x - 9) {(x + 3)}^{- 1}$

$x + 3 x^{3} - 11 x^{2} + 48 x - 167 x^{4} - 8 x^{3} + 15 x^{2} - 23 x - 9 - (x^{4} + 3 x^{3}) - 11 x^{3} + 15 x^{2} - (- 11 x^{3} - 33 x^{2}) 48 x^{2} - 23 x - (48 x^{2} + 144 x) - 167 x - 9 - (- 167 x - 501) 492 + \frac{492}{x + 3}$

$x^{3} - 11 x^{2} + 48 x - 167 + \frac{492}{x + 3}$

$\frac{36 x^{2} y^{3} + 12 x y^{2} - 24 x y}{- 9 x y}$

$\frac{36 x^{2} y^{3}}{- 9 x y} + \frac{12 x y^{2}}{- 9 x y} - \frac{24 x y}{- 9 x y}$

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

$\frac{3 x^{3} + 5 x^{2} - 2 x + 1}{x - 1}$

$x - 1 3 x^{2} + 8 x + 6 3 x^{3} + 5 x^{2} - 2 x + 1 - (3 x^{3} - 3 x^{2}) 8 x^{2} - 2 x - (8 x^{2} - 8 x) 6 x + 1 - (6 x - 6) 7 + \frac{7}{x - 1}$

$3 x^{2} + 8 x + 6 + \frac{7}{x - 1}$

$(2 y^{3} - 6 y - 4) \div (y + 1)$

$y + 1 2 y^{2} - 2 y - 4 2 y^{3} + 0 y^{2} - 6 y - 4 - (2 y^{3} + 2 y^{2}) - 2 y^{2} - 6 y - (- 2 y^{2} - 2 y) - 4 y - 4 - (- 4 y - 4) 0$

$2 y^{2} - 2 y - 4$

$\frac{y^{4} - 3 y^{3} + 3 y^{2} + 72 y - 22}{y^{2} + 3 y - 1}$

$y^{2} + 3 y - 1 y^{2} - 6 y + 22 y^{4} - 3 y^{3} + 3 y^{2} + 72 y - 22 - (y^{4} + 3 y^{3} - y^{2}) - 6 y^{3} + 4 y^{2} + 72 y - (- 6 y^{3} - 18 y^{2} + 6 y) 22 y^{2} + 66 y - 22 - (22 y^{2} + 66 y - 22) 0$

$y^{2} - 6 y + 22$

$(14 a^{4} b^{3} - 28 a^{2} b^{2} + 35 a b) {(7 a^{2} b)}^{- 1}$

$\frac{14 a^{4} b^{3} - 28 a^{2} b^{2} + 35 a b}{7 a^{2} b} \frac{14 a^{4} b^{3}}{7 a^{2} b} - \frac{28 a^{2} b^{2}}{7 a^{2} b} + \frac{35 a b}{7 a^{2} b}$

$2 a^{2} b^{2} - 4 b + \frac{5}{a}$

$(5 r^{3} w + 9 r^{2} w^{2} - 3 r w^{3}) \div 45 r$

$\frac{5 r^{3} w}{45 r} + \frac{9 r^{2} w^{2}}{45 r} - \frac{3 r w^{3}}{45 r}$

$\frac{r^{2} w}{9} + \frac{r w^{2}}{5} - \frac{w^{3}}{15}$

$5 x - 1 5 x^{3} + 14 x^{2} - 53 x + 14$

$5 x - 1 x^{2} + 3 x - 10 5 x^{3} + 14 x^{2} - 53 x + 14 - (5 x^{3} - x^{2}) 15 x^{2} - 53 x - (15 x^{2} - 3 x) - 50 x + 14 - (- 50 x + 10) 4 + \frac{4}{5 x - 1}$

$x^{2} + 3 x - 10 + \frac{4}{5 x - 1}$

$\frac{3 x^{4} + 17 x^{3} - 12 x^{2} - 19 x + 7}{3 x^{2} + 2 x - 1}$

$3 x^{2} + 2 x - 1 x^{2} + 5 x - 7 3 x^{4} + 17 x^{3} - 12 x^{2} - 19 x + 7 - (3 x^{4} + 2 x^{3} - x^{2}) 15 x^{3} - 11 x^{2} - 19 x - (15 x^{3} + 10 x^{2} - 5 x) - 21 x^{2} - 14 x + 7 - (- 21 x^{2} - 14 x + 7) 0$

$x^{2} + 5 x - 7$

$(2 x^{4} - 3 x^{3} + x - 2) \div (x - 2)$

$x - 2 2 x^{3} + x^{2} + 2 x + 5 2 x^{4} - 3 x^{3} + 0 x^{2} + x - 2 - (2 x^{4} - 4 x^{3}) x^{3} + 0 x^{2} - (x^{3} - 2 x^{2}) 2 x^{2} + x - (2 x^{2} - 4 x) 5 x - 2 - (5 x - 10) 8 + \frac{8}{x - 2}$

$2 x^{3} + x^{2} + 2 x + 5 + \frac{8}{x - 2}$

$(27 y^{3} - 8) {(3 y - 2)}^{- 1}$

$3 y - 2 9 y^{2} + 6 y + 4 27 y^{3} + 0 y^{2} + 0 y - 8 - (27 y^{3} - 18 y^{2}) 18 y^{2} + 0 y - (18 y^{2} - 12 y) 12 y - 8 - (12 y - 8) 0$

$9 y^{2} + 6 y + 4$

$(10 g^{5} h^{4} - 12 g^{4} h^{3} + 15 g^{3} h + 5 g h^{2}) \div (- 15 g^{2} h)$

$\frac{10 g^{5} h^{4}}{- 15 g^{2} h} - \frac{12 g^{4} h^{3}}{- 15 g^{2} h} + \frac{15 g^{3} h}{- 15 g^{2} h} + \frac{5 g h^{2}}{- 15 g^{2} h}$

$- \frac{2 g^{3} h^{3}}{3} + \frac{4 g^{2} h^{2}}{5} - g - \frac{h}{3 g}$

Note

Your student may also write this as

$- (\frac{2 g^{3} h^{3}}{3} - \frac{4 g^{2} h^{2}}{5} + g + \frac{h}{3 g})$

Determine the height of the pyramid when the volume is $6 x^{3} + 19 x^{2} - 11 x - 14$ cubic inches and an area of the base is $2 x^{2} + 5 x - 7$ square inches.

$V = \frac{1}{3} l w h; Area of the base = l w 3 V = l w h 3 (6 x^{3} + 19 x^{2} - 11 x - 14) = (2 x^{2} + 5 x - 7) h h = \frac{18 x^{3} + 57 x^{2} - 33 x - 42}{2 x^{2} + 5 x - 7}$

$2 x^{2} + 5 x - 7 9 x + 6 18 x^{3} + 57 x^{2} - 33 x - 42 - (18 x^{3} + 45 x^{2} - 63 x) 12 x^{2} + 30 x - 42 - (12 x^{2} + 30 x - 42) 0$

The height of the pyramid is 9x + 6 inches.

The area of a rectangle is $5 x^{2} + 13 x + 6$ square feet. Find the width of the rectangle if the length is 5x + 3 feet.

$A = l w 5 x^{2} + 13 x + 6 = (5 x + 3) w w = \frac{5 x^{2} + 13 x + 6}{5 x + 3} 5 x + 3 x + 2 5 x^{2} + 13 x + 6 - (5 x^{2} + 3 x) 10 x + 6 - (10 x + 6) 0$

The width is x + 2 feet.