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MATHEMATICS 8 SECOND QUARTER EXAMINATION NAME:_________________________
Date:________________
Score:_____________
I. Multiple Choice: Identify the choice that best completes completes the statement statement or answers answers the question. ( Each item item is worth two points, points, show your your solution for each item on the the right side side of this paper to get the full points.) points.)
______1.
Simplify
.
A.
C.
B.
D.
______2. A. B.
Multiply.
______3. A. B.
Multiply.
______4. A. B.
Multiply.
______5. A. B.
Factor
______6. A.
Factor the polynomial polynomial
C. D.
C. D.
C. D. . C. D. .
B. Cannot be factored C. D. ______7.
Simplify the the rational rational expression
A. 3r ; r B. 3r ; r
3 3 or 0
. Identify any any excluded excluded values. values. C. 3r ; no excluded values D. ; r 3
______8.
Simplify the rational expression
.
A.
C.
B.
D.
______9.
Simplify the rational expression
.
A.
C.
B.
D.
______10.
Multiply. Simplify your answer.
A.
C.
B.
D.
______11.
Multiply. Simplify your answer.
A.
C.
B.
D.
______12.
Multiply. Simplify your answer.
.
A.
C.
B.
D.
______13.
Divide. Simplify your answer.
A.
C.
B.
D.
______14.
Divide
. Assume that all expressions are defined.
A.
C.
B.
D.
______15.
Add. Simplify your answer.
A.
C.
B.
D.
______16.
Subtract. Simplify your answer.
A.
C.
B.
D.
______17.
Add. Simplify your answer.
A.
C.
B.
D.
______18. A.
Subtract and simplify. Find the excluded values. ;
B. C. D.
______19.
Add
;
and
;
,
;
and
and
.
A.
C.
B.
D.
______20.
Simplify
. Assume that all expressions are defined.
A.
C.
B.
D.
II. Arrange the steps in adding/subtracting rational expressions in order by writing the numbers 1 to 5, 1 being the first step to be done and 5 being the last, on the space provided.
_____ Add or subtract the numerators. _____ Simplify, if possible. _____ If necessary, find equivalent fractions that have the same denominator. _____ Factor, if possible. _____ Combine any like terms in the numerator. III.
Discussion.
1. The area of a rectangle is equal to square units. If the length of the rectangle is equal to how will you find the expression that represents its width? What is this expression?
units,
2. The amount of paint needed to cover a wall is proportional to its area. The wall is rectangular and has an area of (4z3 + 8z2 + 4z) square meters. A. List all possible expressions for the length and width of the wall. (Assume the factors are polynomials.) B. Give a value for z to find each possible length and width.
3. The formula for the difference in volumes between two square prisms of equal height is , where h is the height of the prisms, S is the base length of the larger prism, and s is the base length of the smaller prism. Write the formula for the difference in volumes between two square prisms in factored form. Then, use the factored form to find the difference in volumes with cm, cm, and cm. s S
h h
2ND QUARTER EXAM MATHEMATICS 8 Answer Section MULTIPLE CHOICE
1. ANS: B =
Rewrite
without negative or zero
exponents. Simplify each part of the expression. =
.
=
.
=
Feedback A
B C
D
Any number to the zero power is equal to 1. A negative exponent in the numerator becomes positive in the denominator. Correct! Any number to the zero power is equal to 1. A negative exponent in the numerator becomes positive in the denominator. A negative exponent in the denominator becomes positive in the numerator.
PTS: OBJ: NAT: 2. ANS:
1 DIF: Advanced REF: Page 396 7-1.4 Simplify Expressions with Zero and Negative Exponents 12.5.3.d TOP: 7-1 Integer Exponents KEY: exponent | negative exponent | simplify C Use FOIL. Distribute n and – 5. Distribute n and – 5 again. Multiply. Combine like-terms.
Feedback A B C D
Distribute again, multiply and combine like-terms. You did not multiply the outer terms. Correct! You did not multiply the inner and outer terms.
PTS: 1 NAT: 12.5.3.c 3. ANS: B
DIF: Basic REF: Page 448 TOP: 7-8 Multiplying Polynomials
OBJ: 7-8.3 Multiplying Binomials KEY: binomial | multiplication
Distribute
and
Distribute
and
Multiply. Combine like terms.
. again.
Feedback A B C D
Check the signs. Correct! Combine only like terms. Combine only like terms.
PTS: 1 NAT: 12.5.3.c 4. ANS: A
DIF: Average REF: Page 449 TOP: 7-8 Multiplying Polynomials
=
Use the rule for
OBJ: 7-8.4 Multiplying Polynomials
.
Use the FOIL method, and then combine like terms. Simplify.
Feedback A B C D
Correct! The terms in the product should be squares. First, use the FOIL method. Then, combine the like terms. Use the FOIL method.
PTS: 1 DIF: Basic REF: Page 457 OBJ: 7-9.3 Finding Products in the Form (a + b)(a – b) NAT: 12.5.3.c TOP: 7-9 Special Products of Binomials KEY: binomial | multiplication 5. ANS: D Since a = 2, the coefficients of the First terms must be factors of 2. Since c = 6, the Last terms must be factors of 6. Since b = , the Outer and Inner products must add up to . The sum of the products of the outer and inner terms should be . It may be helpful to make a table to check all the factors of 2 and all the factors of 6. Then check the products of the outer and inner terms to see if the sum is . Feedback A B
C D
You reversed the second terms in the parentheses. When b is negative, the factors of c are both negative. When b is positive, the factors of c are both positive. The coefficient of the x-term in the second binomial cannot be 1. Check your answer. Correct!
PTS: OBJ: TOP: 6. ANS:
1 DIF: Basic REF: Page 506 8-4.2 Factoring ax^2 + bx + c When c Is Positive 8-4 Factoring ax^2 + bx + c A
NAT: 12.5.3.d
Find the GCF. The GCF of , , and Write the terms as products using the GCF.
is
Use the Distributive Property to factor out the GCF.
Feedback A B C
Correct! First, find the GCF. Then, use the Distributive Property to factor out the GCF. First, find the GCF. Then, write the terms as products using the GCF.
D
First, find the GCF. Then, write the terms as products using the GCF.
PTS: 1 DIF: Average REF: Page 487 OBJ: 8-2.1 Factoring by Using the GCF NAT: 12.1.5.b TOP: 8-2 Factoring by GCF 7. ANS: A Factor common factors out of the numerator and/or denominator. Divide out the common factors to simplify the expression. Finally, use the original denominator to determine excluded factors. Feedback A B C D
Correct! Determine excluded values from the original denominator. Determine excluded values from the original denominator. Divide out common factors.
PTS: 1 NAT: 12.5.3.c 8. ANS: B
DIF: Basic REF: Page 643 OBJ: 10-3.2 Simplifying Rational Expressions TOP: 10-3 Simplifying Rational Expressions
Factor the numerator and denominator. Divide out the common factors. Simplify.
Feedback A B C D
Factor the denominator. Divide out common factors. Correct! Factor the denominator. Divide out common factors. Factor the denominator. Divide out common factors.
PTS: OBJ: NAT: 9. ANS:
1 DIF: Basic REF: Page 643 10-3.3 Simplifying Rational Expressions with Trinomials 12.5.3.c TOP: 10-3 Simplifying Rational Expressions B
Factor.
=
Identify the opposite binomials.
=
Divide out the common factors, and simplify.
=
Feedback A B C D
Factor the denominator also. Correct! Identify the opposite binomials. Then, divide out common factors and simplify. Identify the opposite binomials. Then, divide out common factors and simplify.
PTS: 1 DIF: Average REF: Page 644 OBJ: 10-3.4 Simplifying Rational Expressions Using Opposite Binomials
NAT: 12.5.3.c 10. ANS: C
TOP: 10-3 Simplifying Rational Expressions
Arrange the expressions so like terms are together:
.
Multiply the numerators and denominators, remembering to add exponents when multiplying: Divide, remembering to subtract exponents: Since
.
.
, this expression simplifies to
Feedback A B C D
A variable raised to the 0 power simplifies to 1. When dividing powers with the same base, subtract the exponents. Correct! Multiply, then simplify.
PTS: 1 NAT: 12.5.3.c 11. ANS: B
DIF: Average REF: Page 652 OBJ: 10-4.1 Multiplying Rational Expressions TOP: 10-4 Multiplying and Dividing Rational Expressions
=
Write the polynomial over 1.
=
Factor the numerator and denominator.
=
Divide out common factors.
=
Multiply remaining factors.
Feedback A B C D
Divide out common factors, and multiply the remaining factors. Correct! First, rewrite the polynomial as an expression over 1. Then, multiply and simplify. Continue simplifying until the answer is in simplest form.
PTS: OBJ: NAT: 12. ANS:
1 DIF: Average REF: Page 653 10-4.2 Multiplying a Rational Expression by a Polynomial 12.5.3.c TOP: 10-4 Multiplying and Dividing Rational Expressions A
=
Factor the numerator and denominator.
=
Simplify.
=
Multiply the remaining factors.
Feedback A
Correct!
B C D
Factor the numerator and denominator and divide out the common factors. Factor the numerator and denominator and divide out the common factors. Factor the numerator and denominator and divide out the common factors.
PTS: OBJ: NAT: 13. ANS:
1 DIF: Average REF: Page 653 10-4.3 Multiplying Rational Expressions Containing Polynomials 12.5.3.c TOP: 10-4 Multiplying and Dividing Rational Expressions A
Write as multiplication by the reciprocal.
=
Multiply the numerators and the denominators.
=
Divide out common factors. Simplify.
=
Feedback A B C
D
Correct! Divide out common factors, and simplify. First, write as multiplication by the reciprocal. Then, multiply the numerators and the denominators. Write as multiplication by the reciprocal first.
PTS: OBJ: NAT: 14. ANS:
1 DIF: Basic REF: Page 654 10-4.4 Dividing by Rational Expressions and Polynomials 12.5.3.c TOP: 10-4 Multiplying and Dividing Rational Expressions A Rewrite as multiplication by the reciprocal.
Simplify by canceling common factors.
= =
Feedback A B C D
Correct! To divide by a fraction, you multiply by its reciprocal. To divide by a fraction, you multiply by its reciprocal. Multiply the first fraction by the reciprocal of the second fraction.
PTS: NAT: KEY: 15. ANS:
1 DIF: Basic REF: Page 579 OBJ: 8-2.4 Dividing Rational Expressions 12.5.3.c TOP: 8-2 Multiplying and Dividing Rational Expressions divide rational expressions | simplify C
Combine like terms in the numerator.
Factor. Divide out common factors. Simplify.
Feedback A B C D
Divide out only common factors. Factor the denominator and continue simplifying your answer. Correct! The denominators are the same, so keep the common denominator.
PTS: OBJ: NAT: 16. ANS:
1 DIF: Average REF: Page 659 10-5.1 Adding Rational Expressions with Like Denominators 12.5.3.c TOP: 10-5 Adding and Subtracting Rational Expressions D
= =
Subtract the numerators.
=
Combine like terms.
=
Factor. Divide out common factors.
=
Simplify.
Feedback A B C D
Factor the numerator and the denominator. Then, divide out common factors. Subtract, not add. Subtract the numerator. Then, combine like terms. Correct!
PTS: OBJ: NAT: 17. ANS:
1 DIF: Average REF: Page 660 10-5.2 Subtracting Rational Expressions with Like Denominators 12.5.3.c TOP: 10-5 Adding and Subtracting Rational Expressions C Identify the LCD,
.
= =
Rewrite each fraction with a denominator of
=
Add.
.
=
and are not like terms, so they cannot be combined. Factor and divide out common factors.
=
Simplify.
Feedback A B C D
To simplify the numerator, factor out 3 y. First find a common denominator. Then add the numerators. Correct! To simplify the numerator, factor out 3 y.
PTS: OBJ: NAT: 18. ANS:
1 DIF: Basic REF: Page 661 10-5.4 Adding and Subtracting with Unlike Denominators 12.5.3.c TOP: 10-5 Adding and Subtracting Rational Expressions B
Factor the difference of squares.
=
Write each expression using the LCD.
=
Subtract the numerators.
=
Distribute.
=
Simplify.
=
Excluded values are values that make the denominator equal to 0. If Therefore, and .
or
, then the denominator is 0.
Feedback A
B C D
First, factor the difference of squares. Then, write each expression using the LCD, subtract, and simplify. Correct! Excluded values are values that make the denominator equal to 0. Check the step where you used the Distributive Property.
PTS: 1 DIF: Advanced NAT: 12.5.3.c TOP: 10-5 Adding and Subtracting Rational Expressions 19. ANS: D
KEY: rational expression
Factor the denominators. The LCD is =
Multiply by
= =
Add the numerators.
.
.
=
Factor the numerator.
=
Divide the common factor.
Feedback A B
C D
Find a common denominator before adding the fractions. This is the first fraction rewritten with the common denominator. Add this to the second fraction. Find a common denominator before adding the fractions. Correct!
PTS: 1 DIF: Average REF: Page 584 OBJ: 8-3.3 Adding Rational Expressions NAT: 12.5.3.c TOP: 8-3 Adding and Subtracting Rational Expressions KEY: add rational expressions 20. ANS: C Method 1 Write the complex fraction as division. Divide. =
Multiply by the reciprocal.
=
Add by finding the LCD.
=
The common factor (
=
Simplify.
) cancels.
Method 2 Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator.
The LCD is
.
=
Cancel common factors.
=
Simplify.
Feedback A
B C D
Multiply both addends in the numerator by the reciprocal of the denominator, not just the first. Multiply the numerator by the reciprocal of the denominator. Correct! Multiply the numerator by the reciprocal of the denominator.
PTS: 1 DIF: Average REF: Page 586 OBJ: 8-3.5 Simplifying Complex Fractions NAT: 12.5.3.c TOP: 8-3 Adding and Subtracting Rational Expressions KEY: simplify complex fractions
21. ANS: B Area of a rectangle equals length times width. Substitute the area and length expressions given. Factor and solve for w. Simplify.
Feedback A
B C D
Multiply the expressions for length and width to be sure they create the polynomial expression for area. Correct! Factor the polynomial to find the possible length and width. In a rectangle, length times width equals area.
PTS: 1 DIF: Advanced NAT: 12.5.3.c TOP: 8-2 Multiplying and Dividing Rational Expressions 22. ANS: A The GCF is 2 z . A = A = 2 z (2 z ) + 2 z (1) A = 2 z (2 z + 1)
KEY: areas | divide rational expressions
Write each term as a product with the GCF as a factor. Use the Distributive Property to factor out the GCF.
Feedback A B C D
Correct! Check your answer by multiplying the factors. Check your answer by multiplying the factors. Check your answer by multiplying the factors.
PTS: 1 NAT: 12.5.3.d 23. ANS: B
DIF: Average REF: Page 488 TOP: 8-2 Factoring by GCF
OBJ: 8-2.2 Application
Factor out the GCF, h. If
cm,
cm, and
cm, then
cm Feedback A B C D
Substitute the values into the equation to find the difference in volumes. Correct! Factor out the GCF, h. Factor out the GCF, h.
PTS: 1
DIF:
Advanced
NAT: 12.5.3.c
TOP: 8-2 Factoring by GCF
