Lesson 6 Puzzling Over Polynomials Practice Understanding

Ready

Divide out all of the common factors. (Assume no denominator equals $0$ .)

1.

$\frac{3 x}{6 x^{2}}$

2.

$\frac{2 \cdot 5 \cdot x \cdot x \cdot x \cdot y}{3 \cdot 5 \cdot x \cdot y \cdot y}$

3.

$\frac{7 a b^{2}}{7 a b^{2}}$

4.

$\frac{(x + 2) (x - 9)}{(x + 2) (x - 9)}$

5.

$\frac{(3 x - 5) (x + 4)}{(x - 1) (3 x - 5)}$

6.

$\frac{(2 x - 11) (3 x + 17)}{(2 x - 11) (3 x - 5)}$

7.

$\frac{(8 x - 7) (x + 3)}{8 x (x + 3) (2 x - 3)}$

8.

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

9.

Why is it important that the instructions say to assume that no denominator equals $0$ ?

Set

Some information has been given for each polynomial. Complete the missing information.

10.

Function: $f (x) = x^{3}$

Function in factored form:

End behavior:

As $x \to - \infty$ , $f (x) \to$ .

As $x \to \infty$ , $f (x) \to$ .

Roots (with multiplicity):

Degree:

Value of leading coefficient:

Graph:

11.

Function in standard form:

Function in factored form: $g (x) = - x (x - 2) (x - 4)$

End behavior:

As $x \to - \infty$ , $g (x) \to$ .

As $x \to \infty$ , $g (x) \to$ .

Roots (with multiplicity):

Degree:

Value of leading coefficient:

Graph:

12.

Function in standard form: $h (x) = x^{3} - 2 x^{2} - 3 x$

Function in factored form:

End behavior:

As $x \to - \infty$ , $h (x) \to$ .

As $x \to \infty$ , $h (x) \to$ .

Roots (with multiplicity):

Degree:

Value of $h (2)$ :

Graph:

13.

Graph:

Function in standard form:

Function in factored form:

End behavior:

As $x \to - \infty$ , $f (x) \to$ .

As $x \to \infty$ , $f (x) \to$ .

Roots (with multiplicity):

Degree:

$y$ -intercept:

14.

Graph:

Function in standard form:

Function in factored form:

End behavior:

As $x \to - \infty$ , $p (x) \to$ .

As $x \to \infty$ , $p (x) \to$ .

Roots (with multiplicity):

Degree:

Value of leading coefficient:

15.

Function in standard form: $q (x) = x^{3} + 2 x^{2} + x + 2$

Function in factored form:

End behavior:

As $x \to - \infty$ , $q (x) \to$ .

As $x \to \infty$ , $q (x) \to$ .

Roots (with multiplicity): $x = i$

Degree:

$y$ -intercept:

Graph:

16.

Finish the graph if it is an even function.

17.

Finish the graph if it is an odd function.

Go

Write the polynomial function in standard form given the leading coefficient and the roots of the function.

18.

Leading coefficient: $2$
Roots: $2$ , $\sqrt{2}$ , $- \sqrt{2}$

19.

Leading coefficient: $- 1$
Roots: $1$ , $1 + \sqrt{3}$ , $1 - \sqrt{3}$

20.

Leading coefficient: $2$
Roots: $4 i$ , $- 4 i$

Fill in the blanks to make a true statement.

21.

If $f (b) = 0$ , then a factor of $f (x)$ must be .

22.

The rate of change in a linear function is always a .

23.

The rate of change of a quadratic function is .

24.

The rate of change of a cubic function is .

25.

The rate of change of a polynomial function of degree $n$ can be described by a function of degree .