Targeted Review Solutions

What is an objective function when working with an optimization problem?

Sample:
The objective function is the function that is used to determine the minimum and maximum for a linear programming problem.

Determine the least common multiple (LCM) for $7 x^{2}, 14 x y, 2 z .$

$LCM (7, 14, 2) = 14 LCM (x^{2}, x y, z) = x^{2} y z$

$14 x^{2} y z$

Simplify.

Note

Problems 3–4
Recall that a fraction bar represents division. The terms can be stacked vertically (problem 3) or written horizontally (problem 4).

$\frac{(\frac{5}{3} - \frac{1}{2})}{2}$

$(\frac{5}{3} - \frac{1}{2}) \div 2 (\frac{10}{6} - \frac{3}{6}) \cdot \frac{1}{2} = \frac{7}{6} \cdot \frac{1}{2}$

$\frac{7}{12}$

$(2 \frac{3}{5}) \div (3 \frac{1}{2})$

$(2 \frac{3}{5}) \div (3 \frac{1}{2}) \frac{13}{5} \div \frac{7}{2} = \frac{13}{5} \cdot \frac{2}{7}$

$\frac{26}{35}$

Factor completely: $4 (x^{2} - 1) + z^{2} (1 - x^{2})$

$4 (x^{2} - 1) - z^{2} (x^{2} - 1) (x^{2} - 1) (4 - z^{2}) (4 - z^{2}) (x^{2} - 1) - (z^{2} - 4) (x^{2} - 1)$

$- (z + 2) (z - 2) (x + 1) (x - 1)$

Simplify: $(x - 3) (x - 5) - (x^{2} - 2 x - 3)$

$x^{2} - 8 x + 15 - x^{2} + 2 x + 3$

$- 6 x + 18$

Bixby’s Bead Shop placed three orders for black, white, and purple beads.
In September, 30 black, 50 white, and 80 purple beads were purchased for $460.
In October, 80 black and 20 white beads were purchased for $260.
In November, $166 was spent on 22 white and 36 purple beads. Write a system with three variables. Do not solve.

$b : black, w : white, p : purple 30 b + 50 w + 80 p = 460 80 b + 20 w = 260 22 w + 36 p = 166$

Determine the value of Q that will make the equation a polynomial identity.
${(Q x - 3)}^{2} = {(2 x - 1)}^{2} - 8 (x - 1)$

${(Q x - 3)}^{2} Q^{2} x^{2} - 3 Q x - 3 Q x + 9 Q^{2} x^{2} - 6 Q x + 9$

${(2 x - 1)}^{2} - 8 (x - 1) 4 x^{2} - 2 x - 2 x + 1 - 8 x + 8 4 x^{2} - 12 x + 9$

$\begin{array}{rcl} - 6 Q x & = & - 12 x \\ 6 Q & = & 12 \\ Q & = & 2 \end{array}$

Multiple Choice

Determine the range of the function when the domain is all real numbers.

all real numbers
y ≤ 1
y ≥ 0
y ≥ 1

The range represents the y-values. The graph approaches the horizontal asymptote, y = 1, and is above this line. This makes the range ≥ 1.

Note

Exponential functions do not have a range of all real numbers.
The y-values are above 1, therefore they cannot be less than 1.
The y-values approach 1, not zero.

Determine the value of (y + z) for the system: $2 x - 3 y - 3 z = 22 2 x + y + z = 14$

16
8
2
–2

$\begin{array}{rcl} (3) (2 x + y + z & = & 14) \Rightarrow \begin{array}{l} 2 x - 3 y - 3 z = 20 \\ + 6 x + 3 y + 3 z = 42 \\ 8 x = 64 \\ x = 8 \end{array} \\ 2 (8) + y + z & = & 14 \\ 16 + y + z & = & 14 \\ y + z & = & - 2 \end{array}$

Note

Your student should solve for x to find the sum of y and z.

This option is the value of 2x. 
This option is the value of x. 
This option is the value when the terms are subtracted in the wrong order.

Determine the expression that when set equal to ${(a x)}^{3} - {(b y)}^{3}$ would form a polynomial identity.

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

This is the polynomial identity for the difference of cubes.

Note

The signs in the second expression are incorrect.

The coefficients a and b also need to be squared.
The middle term in the second expression should not have the coefficient 2.

The formula for the difference of cubes is $(a^{3} - b^{3}) = (a - b) (a^{2} + a b + b^{2})$

Select the word that best represents the polynomial.
An expression with three terms with 2 as the highest degree

linear binomial
linear trinomial
quadratic trinomial
binomial trinomial

Note

A–B) A linear expression has the highest degree of 1.

An expression cannot be a binomial and trinomial at the same time.

Problem	1	2	3	4	5	6	7	8	9	10	11	12
Origin	L01	FD	FD	FD	L03	L03	L02	L04	A1	L02	L04	A1

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