Polynomials
Inequalities
Exponents
Transformation Properties
Similarity
100
(4x^3-2x^2+1)+(7x^2+12x)
A.4x^3+3x^2+6x+1
B.4x^2+5x^3+12x+1
C. 4𝑥^3+5𝑥^2+12𝑥+1
D. 4x^2+3x^2+12x+1
C. 4𝑥^3+5𝑥^2+12𝑥+1
Step 1: Elimination
(4𝑥3−2𝑥2+1)+(7𝑥2+12𝑥)
(4x^{3}-2x^{2}+1)+(7x^{2}+12x)(4x3−2x2+1)+(7x2+12x)
4x^3-2x^2+1+7x^2+12x
Step 2: Combine Like Terms
(4𝑥3−2𝑥2+1)+(7𝑥2+12𝑥)
4𝑥^3+5𝑥^2+1+12𝑥
100
3(x – 1) ≤ 2(x – 3)
A. x<-3
B. x>-3
C. x≤-3
D. x>-3
C. x≤-3
Step 1:
3(x – 1) ≤ 2(x – 3)
3x – 3 ≤ 2x-2x3
3x-3≤2x-6
Step 2:
(3x-3)+(3-2x)≤(2x-6)+(3-2x)
(3x-3)+(-2x+3)≤2x-6)+(-2x+3)
Step 3:
3x-3-2x+3≤2x-6-2x+3
3x-2x-3+3≤2x-2x-6+3
100
What is the equivalent of x cubed (x to the third power)?
A. x plus x plus x
B. x times x times x
C. x minus x minus x
D. x divided by x divided by x
B. x times x times x
X multiplied by itself three times is x times x times x. The third power is just a shortened way to say it to make it less long and it is a shortened way to repeat multiplication. That is why we have to the powers.
100
Identify the transformation from ABC to A'B'C'.
A. 90o clockwise rotation
B. 90o counterclockwise rotation
C. Reflection across the y-axis
D. Reflection across the x-axis
D. Reflection across the x-axis
100
Which of the following explains how △AEB△AEB could be proven similar to △DEC △DEC using the AA similarity postulate?
A. ∠AEB≅∠CED∠AEB≅∠CED because vertical angles are congruent; reflect △ CED △CED across segment FG, then translate point D to point A to confirm ∠EAB≅∠EDC ∠EAB≅∠EDC.
B. ∠AEB≅∠CED ∠AEB≅∠CED because vertical angles are congruent; rotate △CED △CED 180∘ 180∘ around point E, then dilate △CED △CED to confirm ≅≅
C . ∠AEB≅∠DEC ∠AEB≅∠DEC because vertical angles are congruent; rotate △CED △CED 180 180∘ around point E, then translate point D to point A to confirm ∠EAB≅∠EDC ∠EAB≅∠EDC
D. ∠AEB≅∠DEC ∠AEB≅∠DEC because vertical angles are congruent; reflect △CED△CED across segment FG, then dilate △CED△CED to confirm ≅≅
C . ∠AEB≅∠DEC ∠AEB≅∠DEC because vertical angles are congruent; rotate △CED △CED 180 180∘ around point E, then translate point D to point A to confirm ∠EAB≅∠EDC ∠EAB≅∠EDC
Reasoning:
"First, from the given image we will extract the congruent angles:∠AEB≅∠DCE ∠AEB≅∠DCE They are the vertical angles of the triangles postulate AEB AEB and DEC
DEC If we translate 180 180∘ the triangle DEC DEC will be on AEB AEB If we translate point DD to point AA we can confirm the following:∠EAB≅∠EDC ∠EAB≅∠EDC
Thus, it is showing that the triangles AEB AEB and DEC DEC are similar, tested using the similarity postulate AA AA The correct option is C"
200
Write polynomial P as a product of linear factors:P(x)=9x^3−19x−10
A. (x+1)(3x−5)(3x+2)
B. (x+1)(5x-3)(2x+3)
C. (x+1)(3x-5)(2x+3)
D. (x+1)(5x-3)(3x+2)
A. (x+1)(3x−5)(3x+2)
Step 1: Find the zeros
p(1)=9(1)^3−19−10=0
p(−1)=9(−1)^3−19(1)−10=0
Step 2: Common Factor
9x^2−9x−10
=9x^2+6x−15x−10
=3x(3x+2)−5(3x+2)
=(3x−5)(3x+2)
200
5x + 7 < 3(x + 1)
A. x > 2
B. x > 3
C. x > -2
D. x < –2
D. x=-2
Step 1:
5x+7 = 3 x (x+1)
5x+7=3x+3
Step 2:
5x+7=3x+3
5x+7-3x=3x+3-3x
2x+7=3x+3-3x
2x+7=3
Step 3:
2x+7=3
2x+7-7=3-7
2x=3-7
2x=-4
Step 4:
2x=-4
2x/2=-4/2
x=-4/2
200
What is the equivalent of x to the negative two-thirds power?
A. 1 divided by the cube root of x squared
B. Not possible
C. x squared divided by 3
D. 2 divided by the cube root of x
A. 1 divided by the cube root of x squared
Reasoning:
“Fractional exponents allow greater flexibility (you'll see this a lot in calculus), are often easier to write than the equivalent radical format, and permit you to do calculations that you couldn't before.”
200
Identify the transformation from ABCD to A'B'C'D'.
A.Translation
B. Reflection across the y-axis
C. Reflection across the x-axis
D. 90o counterclockwise Rotation
A. Translation
200
Let ABC be a triangle and A'C' a segment parallel to AC. What can you say about triangles ABC and A'BC'? Explain your answer?
Since A'C' is parallel to AC, angles BA'C' and BAC are congruent. Also angles BC'A' and BCA are congruent. Since the two triangles have two corresponding congruent angles, they are similar.
300
What is the leading term of the polynomial 2 x9 + 7 x3 + 191?
A. 3x^7
B. 2x^9
C. 2x^7
D. 3x^9
B. 2x^9
Reasoning:
The leading term of the polynomial is the highest power of x
300
3(x – 1) ≤ 2(x – 3)
A. x ≤ -3
B.x < -3
C. x>-3
D. x ≥ -3
A. x<=-3
Step 1:
3 x (x-1) <=2 x (x-3)
3x-3<=2x(x-3)
3x-3 < = 2x-6
Step 2:
3x-3<=2x-6
3x-3-2x<=2x-6-x
x-3<=2x-6-2x
x-3<=-6
Step 3:
x-3<=-6
x-3+3<=-6+3
x<-6+3
300
What is the equivalent of x squared times x cubed?
A. 5 times x
B. 6 times x
C. x to the sixth power
D. x to the fifth power
D. x to the fifth power
Reasoning:
x^2 x x^3 = (x)(x) x (x)(x)(x) = (x)(x)(x)(x)(x) = x^5
300
Identify the transformation.
A. Reflection across y-axis
B. 90 Rotation counterclockwise
C. Translation 5 units left, 1 unit up.
D. Translation 5 units right, 1 unit down
C. Translation 5 units left, 1 unit up.
300
Show that triangles ABC and A'BC', in the figure below, are similar.
Angles ABC and A'BC' are congruent.
Since the lengths of the sides including the congruent angles are given, let us calculate the ratios of the lengths of the corresponding sides. BA / BA' = 10 / 4 = 5 / 2 BC / BC' = 5 / 2
The two triangles have two sides whose lengths are proportional and a congruent angle included between the two sides. The two triangles are similar.
400
Polynomials should not have negative
A. Terms
B. Coefficients
C. Exponents
D. All of the above
C. Exponents
Reasoning:
You cannot have a negative exponent, you must only have whole number exponents.
400
3x + 9 ≥ –x + 19
A. x > 5/2
B. x ≥ 5/2
C. x < 5/2
D. x ≤ 5/2
B. x ≥ 5/2
Step 1:
3x+9>=-x+19
3x+9+x>=-x+19+x
4x+9>=-x+19+x
4x+9>=19
Step 2:
4x+9>=19
4x+9-9>=19-9
4x>=19-9
4x>=10
Step 3:
4x>=10
x>=10/4
4x/4>=10/4
x>=5/2
400
What is equivalent to x to the one half power?
A. x divided by 2
B. Not possible
C. Square root of x
D.1 divided by x squared
C. Square root of x
Reasoning:
“Taking a number to the power of 12 is the same thing as taking a square root: x1/2=√x.”
400
Choose the correct scale factor:
A. 1/3
B. 2
C. 3
D. 1/2
B. 2
400
In the figure above, lines DG, CF, and BE are parallel. If line segment AB = 6, line segment AE = 9, line segment EF = 10, and line segment FG = 11, what is the length of line AD?
A. 23
B. 24
C. 23
D. 28
E. 20
F. 22
E. 20
Explanation:
A key to solving this problem comes in recognizing that you’re dealing with similar triangles. Because lines BE, CF, and DG are all parallel, that means that the top triangle ABE is similar to two larger triangles, ACF and ADG. You know that because they all share the same angle A, and then if the horizontal lines are all parallel then the bottom two angles of each triangle will be congruent as well. You’ve established similarity through Angle-Angle-Angle.
This means that the side ratios will be the same for each triangle. And for the top triangle, ABE, you know that the ratio of the left side (AB) to right side (AE) is 6 to 9, or a ratio of 2 to 3. Note then that the remainder of the given information provides you the length of the entire right-hand side, line AG, of larger triangle ADG. If AE is 9, EF is 10, and FG is 11, then side AG is 30. And since you know that the left-hand side has a 2:3 ratio to the right, then line segment AD must be 20.
500
x3 + 30x = –13x2
Multiple answers are valid
A. x=0
B. x= -4
C. x= -8
D. x= 6
E. x= -10
F. x= 4
G. x=-3
A E G are all valid answers
A. x=0
E.x=-10
G. x=-3
Step 1:
x^3+13x^2+30x=0
Step 2:
x(x^2+13x+30)=0
x(x+3)(x+10)=0
Step 3:
x= 0
x+3=0
x+10=0
500
3 ( 8 − 4 x ) < 6 ( x − 5 )
A. x > 3
B. x < 3
C. x > -3
D.x < -3
A. x>8
Step 1:
3 x (8-4x) < 6 x (x-5)
24-12x< 6 x (x-5)
24-12x<6x-30
Step 2:
24-12x <6x-30
24-12x-6x<6x-30-6x
-18x+24<6x-30-6x
-18x+24<-30
Step 3:
-18+24 < -30
-18+24-24<-30-24
-18x<-54
-18x<-30-24
Step 4:
-18x<-54
-18x/-18 > -54/-18
x>-54/-18
500
The value of 9 to the power 0 is equal to
A. 1
B. 9
C. 0
D. 1/9
A. 1
Step 1:
9^2 / 9^2 = 9^(2-2) = 9^0 = 1
500
Side AC is about 5.2 units. What is the length of A'C'?
A. 4
B. 5.2
C. 10
D. 8
B. 5.2
500
In the figure above, line segment AC is parallel to line segment BD. If line segment AC = 15, line segment BD = 10, and line segment CE = 30, what is the length of line segment CD?
A. 25
B. 10
C. 15
D. 20
E. 5
B. 10
Explanation:
This problem tests the concept of similar triangles. First, you should recognize that triangle ACE and triangle BDE are similar. You know this because they each have the same angle measures: they share the angle created at point E and they each have a 90-degree angle, so angle CAE must match angle DBE (the top left angle in each triangle.
Because these triangles are similar, their dimensions will be proportional. Since sides, AC and BD - which are proportional sides since they are both across from the same angle, E - share a 3:2 ratio you know that each side of the smaller triangle (BDE) will be ⅔ as long as its counterpart in the larger triangle (ACE). Consequently, if the bottom side CE in the larger triangle measures 30, then the proportional side for the smaller triangle (side DE) will be 23 as long, measuring 20.
Since the question asks for the length of CD, you can take side CE (30) and subtract DE (20) to get the correct answer, 10.