QUESTION 1: Factoring a perfect square trinomial with leading coefficient 1
Factor: y^2 - 12x + 36
EXPLANATION:
When factoring a polynomial, the following formulas are sometimes useful
1.
a^2 + 2ab + b^2 = (a + b)^2
2.
a^2 - 2ab
+ b^2 = (a - b)^2
We can use formula 2. to factor the given perfect square polynomial.
y^2 - 12x + 36 = y^2 – 2(y)(6) + 6^2
= (y – 6)^2
ANSWER: (y –
6)^2
QUESTION 2: Simplifying a ratio of multivariate monomials
Simplify: 25xy / 35yz
EXPLANATION:
We simplify 25xy / 35yz as follows
25xy / 35yz = 5xy / 7yz Canceling the common factor 5
= 5x / 7z Canceling the common factor y
ANSWER: 5x /
7z
QUESTION 3: Factoring a linear binomial
Factor: 5u + 25
EXPLANATION:
We first find the greatest common factor (GCF) of 5u and 25
Note that 5u and 25 have no
common variables
So we just need to find the GCF
of 5 and 25
The GCF is 5
We then factor out the GCF using the distributive property
5u + 25 = 5(u) + 5(5)
= 5(u + 5)
ANSWER: 5(u
+ 5)
QUESTION 4: Cube root of an integer
Find the value of: cube root of
343 or 343^(1/3)
EXPLANATION:
The expression cube root 343 means the same 343^(1/3)
The cube root of 343 is a number
whose cube (3rd power) is 343
Because 343 is positive, its
cube root must also be positive
Checking some positive integers,
we see that 7 is the cube root of 343
1^3 = 1*1*1 = 1
2^3 = 2*2*2 = 8
3^3 = 3*3*3 = 27
4^3 = 4*4*4 = 64
5^3 = 5*5*5 = 125
6^3 = 6*6*6 = 216
7^3 = 7*7*7 = 343
ANSWER: 7
QUESTION 5: Signed fraction subtraction involving double negation
Evaluate and write your answer
in simplest form: -1/6 – (-5/4)
EXPLANATION:
The least common denominator of -1/6 and -5/4 is 12
So we first rewrite both
fractions with a denominator of 12
-1/6 = -1(2) / 6(2) = -2/12
-5/4 = -5(3) / 4(3) = -15/12
Now we can
subtract:
-1/6 – (-5/4) = -2/12 – (-15/12)
= -2/12 + 15/12
= 13/12
ANSWER:
13/12
QUESTION 6: Evaluating expressions with exponents of zero
Evaluate the expressions:
(1/9)^0 = ?
-(4)^0 = ?
EXPLANATION:
For any nonzero number a,
it is always the case that a^0 =1
(1/9)^0 = 1
Because 4^0 = 1, we have –(4)^0
= -1
ANSWER:
(1/9)^0 = 1
-(4)^0 = -1
QUESTION 7: Squaring a binomial
Rewrite without parentheses and
simplify: (y – 4)^2
EXPLANATION:
We can write (y – 4)^2 as (y – 4)(y – 4)
We can multiply (y – 4) by
itself using FOIL (First, Outer, Inner, Last)
F : Multiply the two
First terms: y*y = y^2
O : Multiply the two
Outside terms: y*(-4) = -4y
I : Multiply the two
Inside terms: -4*y = -4y
L : Multiply the two Last
terms: -4*(-4) = 16
The product
(y – 4)(y – 4) is then equal to the sum of these terms.
(y – 4)(y – 4) = y^2 – 4y – 4y + 16
= y^2 – 8y + 16
ANSWER: y^2
– 8y + 16
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