Practice 2 Solutions

Write the polynomial as a product of factors from the graph. Explain the multiplicity.

Note

Problems 1–2

Start by determining the possible degree. You can use the abbreviations C, B, and S on the graph to mark a cross-through, bounce, and snake respectively.

$a < 0, n = odd degree Estimated degree : n = 1 + 1 + 2 + 1 = 5 f (x) = - a (x - (- 4.5)) (x - (- 1.5)) {(x - 0)}^{2} (x - 3.5)$

$f (x) = - a x^{2} (x + 4.5) (x + 1.5) (x - 3.5)$

There is a double root (bounce = 2) when $x = 0$ , a single root (cross = 1) when the graph crosses the x-axis when $x = - 4.5$ , $x = - 1.5$ , and $x = 3.5$ .

$a > 0, n = even degree Estimated degree : n = 1 + 3 + 2 = 6 f (x) = a (x - (- 2)) {(x - 1)}^{3} {(x - 5)}^{2}$

$f (x) = a (x + 2) {(x - 1)}^{3} {(x - 5)}^{2}$

There is a double root (bounce = 2) when $x = 5$ , a single root (cross = 1) when the graph crosses the x-axis when $x = - 2$ , a triple root (snake = 3) when $x = 1$ .

Sketch the roots. Then write the equation.

$a < 0 x = - 5, multiplicity 1 (cross through) x = - 1, multiplict y 2 (bounce) x = 2, multiplicity 3 (snake)$

$n = 1 + 2 + 3 = 6$

$f (x) = - a (x + 5) {(x + 1)}^{2} {(x - 2)}^{3}$

$a > 0 x = - 2, multiplicity 3 (snake)$

$n = 3$

$f (x) = a {(x + 2)}^{3}$

Sketch the zeros of the function.

Note

Problems 5–6

Because answers are sketched, graphs may not look exactly the same as the solutions. The end behavior and how the graph intersects the x-axis is the key to this lesson.

$x^{2} {(x + 2)}^{2} (x - 1) = 0$

$degree : n = 2 + 2 + 1 a > 0, n = 5$

$- {(x - 3)}^{2} {(x + 2)}^{3} (x - 5) = 0$

$degree : n = 2 + 3 + 1 a < 0, n = 6$

List all potential rational roots. Do not solve.

$5 x^{4} - 2 x^{2} + 7 x - 15 = 0$

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 3, \pm 5, \pm 15}{\pm 1, \pm 5} \end{array}$

$\begin{array}{rcl} \pm 1, \pm \frac{1}{5}, \pm 3, \pm \frac{3}{5}, \pm 5, \pm 15 \end{array}$

$g (x) = 4 x^{4} - 6 x^{3} + 9 x^{2} - 13 x + 18$

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18}{\pm 1, \pm 2, \pm 4} \end{array}$

$\begin{array}{rcl} \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 6, \pm 9, \pm \frac{9}{2}, \pm \frac{9}{4}, \pm 18 \end{array}$

Determine all roots of the function.

Note

Problems 9–12

Problems can be solved without using technology.

There are many ways to solve these problems that depend on the order in which you use the possible rational roots.

$3 x^{4} - 10 x^{3} - 24 x^{2} - 6 x + 5 = 0$

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 5}{\pm 1, \pm 3} = \pm 1, \pm \frac{1}{3}, \pm 5, \pm \frac{5}{3} \end{array}$

$3 x^{2} + 2 x - 1 = 0 (3 x - 1) (x + 1) = 0 x = \frac{1}{3}, - 1$

$x = - 1 (mult 2), \frac{1}{3}, 5$

$g (x) = 4 x^{3} - 9 x^{2} + 6 x - 1$

$\begin{array}{rcl} RRT & = & \frac{\pm 1}{\pm 1, \pm 2, \pm 4} = \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4} \end{array}$

$4 x^{2} - 5 x + 1 = 0 (4 x - 1) (x - 1) = 0$

$x = \frac{1}{4}, 1 (mult 2)$

$j (x) = x^{3} + 4 x^{2} + x - 6$

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 2, \pm 3, \pm 6}{\pm 1} = \pm 1, \pm 2, \pm 3, \pm 6 \end{array}$

$x^{2} + 5 x + 6 = 0 (x + 2) (x + 3) = 0 x = - 2, - 3$

$x = - 3, - 2, 1$

$f (x) = x^{4} - 6 x^{3} + 7 x^{2} + 6 x - 8$

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 2, \pm 4, \pm 8}{\pm 1} = \pm 1, \pm 2, \pm 4, \pm 8 \end{array}$

$x^{2} - 6 x + 8 = 0 (x - 4) (x - 2) = 0 x = 4, 2$

$x = - 1, 1, 2, 4$

Determine all roots of the function. Then write them as a product of factors using rational numbers.

Note

Problems 13–16

You do not need to write out all possible rational roots. You are encouraged to use technology for these problems.

$2 x^{3} + 6 x^{2} + 5 x + 2 = 0$

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 2}{\pm 1, \pm 2} = \pm 1, \pm \frac{1}{2}, \pm 2 \end{array}$

$2 x^{2} + 2 x + 1 = 0 x = \frac{- 2 \pm \sqrt{{(2)}^{2} - 4 (2) (1)}}{2 (2)} x = \frac{- 2 \pm \sqrt{- 4}}{2} = \frac{- 2 \pm 2 i}{2} x = - 1 \pm i$

$x = - 1, - 1 \pm i (x + 1) (2 x^{2} + 2 x + 1) = 0$

$x^{4} - 3 x^{3} - 3 x^{2} - 75 x - 700 = 0$

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 2, \pm 4, \pm 5, \pm 7, \pm 10, \pm 14, \pm 20, \pm 25, \pm 28, \pm 35, \pm 50, \pm 70}{\pm 1} \\ = & \frac{\pm 100, \pm 140, \pm 175, \pm 350, \pm 700}{\pm 1} \\ = & \pm 1, \pm 2, \pm 4, \pm 5, \pm 7, \pm 10, \pm 14, \pm 20, \pm 25, \pm 28, \pm 35, \pm 50, \pm 70 \\ \pm 100, \pm 140, \pm 175, \pm 350, \pm 700 \end{array}$

$x^{2} + 25 = 0 x^{2} = - 25 x = \pm 5 i$

$x = - 4, 7, \pm 5 i (x + 4) (x - 7) (x^{2} + 25) = 0$

$r (x) = 16 x^{5} - 32 x^{4} - 81 x + 162$

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54, \pm 81, \pm 162}{\pm 1, \pm 2, \pm 4, \pm 8, \pm 16} \end{array}$

$\begin{array}{rcl} = & \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm \frac{1}{16}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8}, \pm \frac{3}{16}, \pm 6, \pm 9, \\ \pm \frac{9}{2}, \pm \frac{9}{4}, \pm \frac{9}{8}, \pm \frac{9}{16}, \pm 18, \pm \frac{27}{2}, \pm \frac{27}{4}, \pm \frac{27}{8}, \pm \frac{27}{16}, \pm 54, \\ \pm 81, \pm \frac{81}{2}, \pm \frac{81}{4}, \pm \frac{81}{8}, \pm \frac{81}{16}, \pm 162 \end{array}$

$16 x^{2} + 36 = 0 16 x^{2} = - 36 4 x^{2} = - 9 x^{2} = - \frac{9}{4} x = \pm \frac{3}{2} i$

$x = \pm \frac{3}{2}, 2, \pm \frac{3}{2} i r (x) = (2 x - 3) (2 x + 3) (x - 2) (4 x^{2} + 9)$

Note

An alternate way to write the factors is: $(x - \frac{3}{2}) (x + \frac{3}{2}) (x - 2) (16 x^{2} + 36)$

$f (x) = x^{3} - 5 x^{2} + 5 x + 3$

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 3}{\pm 1} = \pm 1, \pm 3 \end{array}$

$x^{2} - 2 x - 1 = 0 x = \frac{- (- 2) \pm \sqrt{{(- 2)}^{2} - 4 (1) (- 1)}}{2 (1)} = \frac{2 \pm 2 \sqrt{2}}{2} x = 1 \pm \sqrt{2}$

$x = 3, 1 \pm \sqrt{2} x = 0.41, 2.41, 3 f (x) = (x - 3) (x^{2} - 2 x - 1)$