Targeted Review Solutions

An Oklahoma license plate has six characters. The first three are letters of the alphabet, while the last three are single-digit numbers. How many different license plates can there be? (Letters and numbers may repeat.)

$26 \cdot 26 \cdot 26 \cdot 10 \cdot 10 \cdot 10$

Oklahoma can have 17,576,000 unique license plates.

A school fair is scheduled, and a committee of three students is needed to welcome guests. If twelve students are available, how many possible groups of students can be formed?

$n C r (12, 3) = \frac{n!}{r! (n - r)!} = \frac{12!}{3! (12 - 3)!} = \frac{12!}{3! 9!} = \frac{12 \cdot 11 \cdot 10 \cdot 9!}{3! 9!} = \frac{2 \cdot 12 \cdot 11 \cdot 10}{3 \cdot 2 \cdot 1}$

There are 220 combinations of students.

Determine the fourth term in the polynomial ${(x - 2 y)}^{5} .$

$\begin{array}{rcl} n C r (5, 3) & = & 10 \\ n & = & 5, r = 3 \\ = & 10 {(x)}^{2} {(- 2 y)}^{3} \\ = & - 80 x^{2} y^{3} \end{array}$

$- 80 x^{2} y^{3}$

Explain what happens to the powers (exponents) of each term as you work through the Binomial Theorem.

Sample: The first term’s exponent starts with n and decreases until it reaches zero. The second term’s exponent starts with zero and increases until it reaches the $n^{th}$ power.

For problems 5–8, use the functions $g (x) = - 6 x + 30, h (x) = x^{2} + 5 .$

$(g \cdot h) (x)$

$g (x) \cdot h (x) (- 6 x + 30) (x^{2} + 5) - 6 x^{3} - 30 x + 30 x^{2} + 150$

$- 6 x^{3} + 30 x^{2} - 30 x + 150$

$g [h (x)]$

$- 6 (x^{2} + 5) + 30 - 6 x^{2} - 30 + 30$

$g [h (x)] = - 6 x^{2}$

$[h \circ g] (x)$

$\begin{array}{rcl} [h \circ g] (x) & = & h [g (x)] \\ = & {(- 6 x + 30)}^{2} + 5 \\ = & (- 6 x + 30) (- 6 x + 30) + 5 \\ = & 36 x^{2} - 180 x - 180 x + 900 + 5 \end{array}$

$\begin{array}{rcl} [h \circ g] (x) & = & 36 x^{2} - 360 x + 905 \end{array}$

$(\frac{h}{g}) (x)$

$\frac{h (x)}{g (x)} = \frac{x^{2} + 5}{- 6 x + 30}$

$\frac{x^{2} + 5}{- 6 (x - 5)}, x \neq 5$

Multiple Choice

Determine the value of $\log_{0.5} 64 .$

–6
8
6
32

$\begin{array}{rcl} \log_{0.5} 64 & = & x \\ 0 . 5^{x} & = & 64 \\ {(\frac{1}{2})}^{x} & = & 2^{6} \\ 2^{- x} & = & 2^{6} \\ - x & = & 6 \\ x & = & - 6 \end{array}$

Note

This option is the square root of 64.
This option is the answer if the negative symbol is ignored.
This option is half of 64.

A marketing firm tested two advertising strategies by showing the ads to 450 users. Afterwards, a random simulation was run 10,000 times.
Group Click Rate Conversion Rate

Treatment: video ad 4.8% 1.2%

Control: static image ad 3.7% 1.3%

The standard deviation for the click rate is 0.45 percentage points. The standard deviation for the conversion rate is 0.13 percentage points.
Determine the z-score for the conversion rate.

0.1
0.77
1.1
2.44

$1.3 - 1.2 = 0.1 z = \frac{0.1}{0.13} = 0.769$

Note

A, C) These options are the observed differences between the treatment and control.

This option is the z-score for the click rate.

A sample of 842 residents of Center County were randomly surveyed about the number of miles to the closest gas station from their homes. Of those surveyed, 64.5% responded “fewer than ten miles from their home.” Find the margin of error to the nearest percentage that most likely contains the population parameter.

It is likely that between 61% and 68% of Center County residents have a gas station fewer than ten miles from their home.
It is likely that between 32% and 39% of Center County residents have a gas station fewer than ten miles from their home.
It is likely that between 62% and 68% of Center County residents have a gas station fewer than ten miles from their home.
Cannot be determined.

$\begin{array}{l} \pm \frac{1}{\sqrt{842}}, n = 842 \\ \pm \frac{1}{\sqrt{842}} = \pm 0.0345 = \pm 3.45 % \\ 64.5 - 3.45 = 61.05 % = 61 % \\ 64.5 + 3.45 = 67.95 % = 68 % \end{array}$

Note

This option is the percentage of residents with a gas station greater than 10 miles from their home.
This option contains a rounding error in which 64.5 is rounded to 65 in the first step.

Select the equation that is NOT a polynomial function.

$f (x) = - 3 x^{5} + 0.5 x^{4} + π x - e$
$g (x) = x^{\frac{1}{3}} (x^{9} - \frac{9}{5} x^{6})$
$h (x) = (12 - x i) (x^{2} + 5)$
$j (x) = 3 x (x - \frac{4}{5}) + \sqrt{6}$

For a polynomial, $p (x) : a_{n} \neq 0,$ n is a whole number, and the coefficients are real numbers.

Note

The coefficients π and e are irrational and real numbers.
When simplified, all exponents are whole numbers.
The square root of a real number can be taken as long as the variable is not under the symbol.

Problem	1	2	3	4	5	6	7	8	9	10	11	12
Origin	L51	L51	L52	L52	L31	L32	L32	L31	L39	L50	L49	L33

L = Lesson in this level, A1 = Algebra 1: Principles of Secondary Mathematics

Group	Click Rate	Conversion Rate
Treatment: video ad	4.8%	1.2%
Control: static image ad	3.7%	1.3%