Name: ________________________ Class: ___________________ Date: __________
03-02 Sample Quiz - Polynomial Identities
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Factor a 2 b 6  x 8 y 2
a.
b.
____
____
(a 2  b 6 )(x 8  y 2 )
(a 4 b 3  xy)(a 4 b 3  xy)
2. Which of the below is a factor of x 6  27
Ê
ˆ
a. ÁÁ x 3  3 ˜˜
Ë
¯
Ê 2
ˆ
Á
b. Á x  3 ˜˜
Ë
¯
c.
d.
c.
d.
(ab 3  x 4 y)(ab 3  x 4 y)
(ab 3 )(b 4  x 4 y)
ÁÊÁ x 2  3x  9 ˜ˆ˜
Ë
¯
ÁÊÁ x 4  3x 2  9 ˜ˆ˜
Ë
¯
3. Vincent is trying to create a polynomial identity.
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Step 0: Original Statement.
(x 2  y 2 ) 2
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Step 1: Expand the binomial expression
(x 2  y 2 )(x 2  y 2 )  x 4  2x 2 y 2  y 4
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Step 2: Add 2x 2 y 2  2x 2 y 2 which is equivalent to adding zero.
x 4  2x 2 y 2  y 4  (2x 2 y 2  2x 2 y 2 )
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Step 3: Rearrange the order of the terms.
x 4  2x 2 y 2  y 4  2x 2 y 2  2x 2 y 2
____________________________________
ˆ2
Ê
Step 4: Factor the first 3 terms x 4  2x 2 y 2  y 4  ÁÁ x 2  y 2 ˜˜ and combine terms.
¯
Ë
2
ÁÊÁ x 2  y 2 ˜ˆ˜  4x 2 y 2
¯
Ë
____________________________________
Step 5: Rewrite the last term as a square.
2
ˆ2
ÊÁ 2
Á x  y 2 ˜˜  ÊÁË 2xy ˆ˜¯
¯
Ë
____________________________________
Are each of his steps valid? If not, in which step is there an error?
a. All of Vincent’s steps are valid.
b. Vincent didn’t correctly expand the binomial in Step 1.
c. Vincent can’t just add to the expression in Step 2.
d. Vincent factored incorrectly the terms in Step 4.
1
ID: A
Name: ________________________
____
4. Factor. 8x 3  125
a. (4x  5)(2x 2  5x  25)
b.
____
(2x  5)(4x 2  10x  25)
ID: A
c.
d.
(8x  5)(x 2  5)
(2x  5)(2x  5)(2x  5)
5. Jessica was trying to prove the polynomial identity:
x2 + y2 = (x+yi) (x – yi)
She made an error because the last step isn’t equivalent.
____________________________________
Step 0: Original Statement.
x 2  y 2  ÊÁË x  yi ˆ˜¯ ÊÁË x  yi ˆ˜¯
____________________________________
Step 1: Expand the RIGHT side using the FOIL technique.
x 2  y 2  x 2  xyi  xyi  y 2
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Step 2: Combine like terms on the RIGHT side.
x2  y2  x2  y2
____________________________________
On which step did she potentially make an algebraic error?
a.
b.
c.
____
Jessica didn’t expand ÊÁË x  yi ˆ˜¯ ÊÁË x  yi ˆ˜¯ correctly in Step 1.
Jessica incorrectly combined the terms in Step 2.
It is impossible to prove because the original identity is false.
6. Which of the below is a binomial factor of the polynomial shown?
4a 2  9
a.
b.
(2a  3i)
(2a  3)
c.
d.
2
(4a  3i)
(a  3)