Practice 2 Solutions

Note

If needed, have your student go back to the Mastery Check and reapply what they have learned to show and say what they know.

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

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

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

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

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

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

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

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

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

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

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

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

$2 y^{2} - 6 y - 4$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.