UNIT 5
Polynomials and Properties of Exponents
LESSON 7
Polynomials and Properties of Exponents Unit Test
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1. What is the degree of the polynomial 3x^4 + 2x^2 - x + 6 ?
a. 4
b. 2
c. 1
d. 3
Answer: a. 4
2. Simplify the expression 5x^3 * 2x^4.
a. 7x^7
b. 10x^7
c. 10x^12
d. 7x^12
Answer: c. 10x^12
3. Which of the following is equivalent to (2x - 3)^2 ?
a. 4x^2 - 6x + 9
b. 4x^2 - 12x + 9
c. 4x^2 - 6x - 9
d. 4x^2 - 12x - 9
Answer: a. 4x^2 - 6x + 9
4. Find the product of (3x - 2) and (x + 4).
a. 3x^2 + 12x - 2x - 8
b. 3x^2 - 2x + 12x - 8
c. 3x^2 + 4x - 8
d. 3x^2 + 4x + 8
Answer: d. 3x^2 + 4x + 8
5. What is the sum of the cubes of the roots of the polynomial x^3 - 8x^2 + 3x - 10?
a. -125
b. -153
c. -243
d. -269
Answer: c. -243
Add or subtract.
(m^2 – m – 4) + (m – 5)
A. m^2 – 2m + 9
B. m^2 + 2m – 9
C. m^2 – 2m – 9
D. m^2 – 9
The solution is:
(m^2 – m – 4) + (m – 5) = m^2 + (-m + m) + (-4 - 5)
Simplifying and combining like terms, we get:
= m^2 - 9
Therefore, the answer is D. m^2 - 9.
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Add or subtract.
(4x^2 – 2x – 1) – (–3x^3 + 2)
A. –3x^3 + 4x^2 – 2x + 1
B. –3x^3 + 4x^2 – 2x – 3
C. 3x^3 + 4x^2 – 2x + 1
D. 3x^3 + 4x^2 – 2x – 3
The solution is:
(4x^2 – 2x – 1) – (–3x^3 + 2) = 4x^2 – 2x – 1 + 3x^3 – 2
Simplifying and combining like terms, we get:
= 3x^3 + 4x^2 - 2x - 3
Therefore, the answer is D. 3x^3 + 4x^2 - 2x - 3.
Suppose you earned 7t – 1 dollars on Monday and 8t + 5 dollars on Tuesday. What were your total earnings? Simplify your answer.
A. –t + 4 dollars
B. –t – 6 dollars
C. 15t – 6 dollars
D. 15t + 4 dollars
The solution is:
Total earnings = earnings on Monday + earnings on Tuesday
= (7t - 1) + (8t + 5)
= 7t + 8t - 1 + 5
Simplifying and combining like terms, we get:
= 15t + 4
Therefore, the answer is D. 15t + 4 dollars.
f^7 ∙ f^1
A. f^8
B. f^7
C. (2f)^8
D. (2f)^7
The solution is:
f^7 ∙ f^1 = f^(7 + 1) = f^8
Therefore, the answer is A. f^8.
Start Fraction 100 superscript 12 baseline over 100 superscript 8 baseline End Fraction
A. 100 superscript Start Fraction 12 over 8 End Fraction baseline
B. 10096
C. 1004
D. 10020
The solution is:
100^12/100^8 = 100^(12-8) = 100^4
Therefore, the answer is C. 1004.
Start Fraction x superscript 13 baseline over x squared End Fraction
A. x superscript thirteen-halves baseline
B. x^15
C. x^11
D. x^26
The solution is:
x^13/x^2 = x^(13-2) = x^11
Therefore, the answer is C. x^11.
Which of the following expressions is true?
A. 4^3 • 4^5 < 4^12
B. 5^2 • 5^3 > 5^5
C. 3^2 • 3^4 = 3^2
D. 5^2 • 5^4 = 5^8
The solution is:
Using the properties of exponents,
A. 4^3 • 4^5 = 4^8 and 4^8 < 4^12, so expression A is true.
B. 5^2 • 5^3 = 5^5 and 5^5 is not greater than 5^5, so expression B is false.
C. 3^2 • 3^4 = 3^6 and 3^2 is not equal to 3^6, so expression C is false.
D. 5^2 • 5^4 = 5^6 and 5^6 is not equal to 5^8, so expression D is false.
Therefore, the answer is A. 4^3 • 4^5 < 4^12.
Which of the following expressions is true?
A. 8^3 • 8^2 < 8^4
B. 4^4 • 4^4 = 4^16
C. 2^2 • 2^6 < 2^8
D. 3^3 • 3^3 > 3^5
The solution is:
Using the properties of exponents,
A. 8^3 • 8^2 = 8^5 and 8^5 is not less than 8^4, so expression A is false.
B. 4^4 • 4^4 = 4^8 and 4^8 is not equal to 4^16, so expression B is false.
C. 2^2 • 2^6 = 2^8 and 2^8 is equal to 2^8, so expression C is true.
D. 3^3 • 3^3 = 3^6 and 3^6 is not greater than 3^5, so expression D is false.
Therefore, the answer is C. 2^2 • 2^6 < 2^8.
Write the value of the expression.
Start Fraction 4 superscript 5 baseline over 4 superscript 5 baseline End Fraction
A. 1
B. 4
C. 5
D. 0
The solution is:
4^5/4^5 = 1
Therefore, the answer is A. 1.
Start Fraction 2 squared over 2 superscript 5 baseline End Fraction
A. 8
B. 6
C. one-eighth
D. –8
The solution is:
2^2/2^5 = 1/2^3 = 1/8
Therefore, the answer is C. one-eighth.
Multiply. Write the result in scientific notation.
(2.3 • 10^1)(7 • 10^6)
A. 1.61 • 10^7
B. 1.61 • 10^8
C. 9.3 • 10^6
D. 9.3 • 10^7
The solution is:
(2.3 • 10^1)(7 • 10^6) = (2.3 * 7) * 10^(1+6) = 16.1 * 10^7
Expressing this in scientific notation format, we get:
= 1.61 * 10^8
Therefore, the answer is B. 1.61 * 10^8.
(1.1 ∙ 10–^5)(3 ∙ 10–^2)
A. 4.1 ∙ 10–^7
B. 4.1 ∙ 10^10
C. 3.3 ∙ 10–^7
D. 3.3 ∙ 10^10
The solution is:
(1.1 ∙ 10^–5)(3 ∙ 10^–2) = (1.1 * 3) * 10^(-5 - 2) = 3.3 * 10^(-7)
Expressing this in scientific notation format, we get:
= 3.3 ∙ 10–^7
Therefore, the answer is C. 3.3 ∙ 10–^7.
Simplify the expression.
8t^5 ∙ 8t^5
A. 64t^25
B. 64t^10
C. 16t^10
D. 16t^5
The solution is:
8t^5 ∙ 8t^5 = (8 ∙ 8) * t^(5 + 5) = 64t^10
Therefore, the answer is B. 64t^10.
Simplify the expression.
–x(6x – 7)
A. –6x^2 + 7x
B. 5x^2 – 6x
C. 6x + 7x
D. –6x – 7x
The solution is:
-x(6x – 7) = -6x^2 + 7x
Therefore, the answer is A. –6x^2 + 7x.
Simplify the expression.
5k^2(–6k^2 – 2k + 6)
A. –30k^3 + 3k^2 + 30k
B. 30k^4 – 10k^3 + 11k^2
C. –k^4 + 3k^3 + 11k^2
D. –30k^4 – 10k^3 + 30k^2
The solution is:
5k^2(–6k^2 – 2k + 6) = -30k^4 - 10k^3 + 30k^2
Therefore, the answer is D. –30k^4 – 10k^3 + 30k^2.
Simplify the expression.
(3k + 2)(k – 3)
A. 3k^2 – 7k – 5
B. 3k^2 – 7k – 6
C. 3k^2 – 4k – 6
D. 3k^2 – 4k – 5
The solution is:
(3k + 2)(k – 3) = 3k^2 -9k + 2k - 6
Simplifying and combining like terms, we get:
= 3k^2 - 7k - 6
Therefore, the answer is B. 3k^2 - 7k - 6.
Simplify the expression.
(–y + 4)(2y – 1)
A. 2y^2 + 7y + 3
B. 2y^2 + 5y – 4
C. –2y^2 + 9y – 4
D. –2y^2 + 6y + 3
The solution is:
(–y + 4)(2y – 1) = -2y^2 + y + 8y - 4
Simplifying and combining like terms, we get:
= -2y^2 + 9y - 4
Therefore, the answer is C. –2y^2 + 9y – 4.
Short Answer
Note: For questions 22–23, your teacher will grade your response to ensure you receive proper credit for your answer.
Look at the given triangles.
triangles
The blue triangle is a right triangle. The vertical leg is labeled with the expression 4 x plus 2. The horizontal leg is labeled with the expression 5 x minus 4. The hypotenuse is labeled with the expression 7 x plus 7.
The red triangle is a right triangle. The vertical leg is labeled with the expression x plus 3. The horizontal leg is labeled with the expression x plus 7. The hypotenuse is labeled with the expression 2 x minus 5.
a. Write an expression in simplest form for the perimeter of each triangle.
b. Write another expression in simplest form that shows the difference between the perimeter of the larger triangle and the perimeter of the smaller triangle.
c. Find the perimeter for each triangle when x = 3
a. The expression for the perimeter of the blue triangle is:
Perimeter = (4x + 2) + (5x - 4) + (7x + 7) = 16x + 5
The expression for the perimeter of the red triangle is:
Perimeter = (x + 3) + (x + 7) + (2x - 5) = 4x + 5
b. The difference between the perimeter of the larger triangle (blue triangle) and the perimeter of the smaller triangle (red triangle) is:
(16x + 5) - (4x + 5) = 12x
So, the expression for the difference is: 12x
c. When x = 3, we can substitute the value of x in the expressions we obtained in part (a) to find the perimeter of each triangle:
Perimeter of blue triangle = 16x + 5 = 16(3) + 5 = 53
Perimeter of red triangle = 4x + 5 = 4(3) + 5 = 17
Therefore, the perimeter of the blue triangle is 53 units and the perimeter of the red triangle is 17 units when x = 3.
Emma, Erin, and Eden completed the problem to the right.
a. Who completed the problem correctly?
b. What did the other two students do wrong in their answers?
Emma's equation shows 6 squared times 6 superscript 5 baseline equals 36 superscript 7 baseline.
Erin's equation shows 6 squared times 6 superscript 5 baseline equals 6 superscript 10 baseline.
Eden's equation shows 6 squared times 6 superscript 5 baseline equals 6 superscript 7 baseline.
a. Emma completed the problem correctly.
b. Erin multiplied 6^2 and 6^5 incorrectly. Instead of adding the exponents, she multiplied them and got 6^10, which is incorrect.
Eden also made a mistake in multiplying 6^2 and 6^5. Instead of adding the exponents, she just multiplied them and got 6^7, which is incorrect.
Hmmmm- I see...
Please let me know if you have any questions or need further clarification on the problem.
1. 6g + 8k - 9g + 2k
A. 3g + 10k
B. -2g+ 10k
C. 17gk
D. 10k - 3
6g + 8k - 9g + 2k = (6g - 9g) + (8k + 2k) = -3g + 10k
Therefore, the answer is B. -2g + 10k.
1. 6g + 8k - 9g + 2k
A. 3g + 10k
B. -3g+ 10k
C. 17gk
D. 10k - 3
Correcting my previous response:
6g + 8k - 9g + 2k = (6g - 9g) + (8k + 2k) = -3g + 10k
Therefore, the answer is B. -3g+ 10k.