Lesson 5 Is This the End? Solidify Understanding

Ready

Multiply.

1.

$(x + 5) (x + 5)$

2.

$(x - 3) (x - 3)$

3.

$(a + b) (a + b)$

4.

In problems 1–3, the answers are called perfect square trinomials. What are the features of a perfect square trinomial?

5.

$(x + 8) (x - 8)$

6.

$(x + \sqrt{3}) (x - \sqrt{3})$

7.

$(x + b) (x - b)$

8.

The products in problems 5–7 end up being binomials, and they are called the difference of two squares.

a.

What are the features of the difference of two squares?

b.

Why don’t they have a middle term like the answers to problems 1–3?

9.

$(x - 3) (x^{2} + 3 x + 9)$

10.

$(x + 10) (x^{2} - 10 x + 100)$

11.

$(a + b) (a^{2} - a b + b^{2})$

12.

a.

The work in problems 9–11 makes them feel like the answers are going to have a lot of terms. What happens in the work of the problems that makes the answers be binomials?

b.

These answers are called the difference of two cubes (problem #9) and the sum of two cubes (problems 10 and 11). What about these answers makes them be the sum or difference of two cubes?

Set

State the $y$ -intercept, the degree, and the end behavior for each of the given polynomials.

13.

$f (x) = x^{5} + 7 x^{4} - 9 x^{3} + x^{2} - 13 x + 8$

$y$ -intercept:

Degree:

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

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

14.

$g (x) = 3 x^{4} + x^{3} + 5 x^{2} - x - 15$

$y$ -intercept:

Degree:

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

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

15.

$h (x) = - 7 x^{9} + x^{2}$

$y$ -intercept:

Degree:

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

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

16.

$p (x) = 5 x^{2} - 18 x + 4$

$y$ -intercept:

Degree:

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

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

17.

$q (x) = x^{3} - 94 x^{2} - x - 20$

$y$ -intercept:

Degree:

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

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

18.

$y = - 4 x + 12$

$y$ -intercept:

Degree:

$As x \to - \infty, y \to$

$As x \to + \infty, y \to$

19.

Identify each function as even, odd, or neither.

a.

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

A.

even

B.

odd

C.

neither

b.

$f (x) = x^{2}$

A.

even

B.

odd

C.

neither

c.

$f (x) = {(x + 1)}^{2}$

A.

even

B.

odd

C.

neither

d.

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

A.

even

B.

odd

C.

neither

e.

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

A.

even

B.

odd

C.

neither

f.

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

A.

even

B.

odd

C.

neither

Go

Fill in the conjugate of the given expression, then multiply the two conjugates.

20.

$(11 + 4 i) ($ $\underset{―}{}$ $) =$ $\underset{―}{}$

21.

$9 i ($ $\underset{―}{}$ $) =$ $\underset{―}{}$

22.

$(2 - 13 i) ($ $\underset{―}{}$ $) =$ $\underset{―}{}$

23.

$- 4 i ($ $\underset{―}{}$ $) =$ $\underset{―}{}$

24.

$(- 1 - 2 i) ($ $\underset{―}{}$ $) =$ $\underset{―}{}$

Solve for $x$ .

25.

$x^{2} + 10 = 0$

26.

$x^{2} + 7 = 0$

27.

$x^{2} + x + 1 = 0$

28.

$x^{2} + x + 2 = 0$