Download Quiz Questions on Exponents, Polynomials, and Radicals from SGPE Summer School 2016 and more Lecture notes Algebra in PDF only on Docsity!
Week 1 Quiz: Basics
SGPE Summer School 2016
Rules for Exponents
Question 1: Simplify the following expression using the various rules for manipulating exponents:
โ^ x^6 x^3 (A) โx^4 (B) x^3 (C)
x^9 (D) x^5.^5 (E) None of the above Answer: (C) โxx^63 = (^) xx 632 = x^6 โ^32 = x^4.^5 = x^92 =
x^9
Question 2: Simplify the following expression using the various rules for manipulating exponents:
x^3 โx
(A) x^32 (B) x^7 (C) โx^4 (D)
x^7 (E) None of the above Answer: (D) x^3 โx = x^3 x^12 = x3+^12 = x^72 = โx^7 Question 3: Simplify the following expression using the various rules for manipulating exponents: (x 4 y^8
(A) (^) yx 117
(B) x y^1224 (C) (x^4 โ y^8 )^3
(D) 3
โx 4 y^8 (E) None of the above
Answer: (B)
(x 4 y^8
= x y^48 โโ 33 = x y^1224
Question 4: Simplify the following expression using the various rules for manipulating exponents:
(x^2 y^2 x^3 y^3 )^16
Answer: (x^2 y^2 x^3 y^3 )^16 = (x2+3y2+3)^16 = (x^5 y^5 )^16 = ((xy)^5 )^16 = (xy)^56 Question 5: Simplify the following expression using the various rules for manipulating exponents:
(xโ (^14) y 3 x^12
)^13
Answer:
(xโ (^14) y 3 x 12
)^13
= (xโ^14 โ^12 y^3 )^13 = (xโ^34 y^3 )^13 = xโ^14 y = (^) xy 14
Polynomials
Question 6: Expand the following expression: (2x + 7)^2 (A) 8x^2 + 18x โ 35 (B) 8x^2 + 49 (C) 4x^2 + 28x + 49 (D) 8x^2 + 18x โ 42 (E) None of the above Answer: (C) (2x + 7)^2 = (2x + 7)(2x + 7) = 4x^2 + 14x + 14x + 49 = 4x^2 + 28x + 49 Question 7: Expand and regroup the following expression: (4x + 3y)(5x^2 โ 2 xy + 6y^2 ) (A) 20x^3 + 7x^2 y + 18xy^2 + 18y^3 (B) 20x^3 โ 2 xy + 18y^3 (C) 9x^3 + 8x^2 y + 6xy^2 + 9y^3 (D) 20x^2 + 25xy + 18y^2 (E) None of the above Answer: (A) (4x + 3y)(5x^2 โ 2 xy + 6y^2 ) = 20x^3 โ 8 x^2 y + 24xy^2 + 15x^2 y โ 6 xy^2 + 18y^3 = 20x^3 + 7x^2 y + 18 xy^2 + 18y^3 Question 8: Expand and regroup the following expression:
(ax + b)^2 + (cx โ d)^2
Answer: (ax + b)^2 + (cx โ d)^2 = (ax + b)(ax + b) + (cx โ d)(cx โ d) = (a^2 x^2 + 2abx + b^2 ) + (c^2 x^2 โ 2 cdx + d^2 ) = (a^2 + c^2 )x^2 + 2(ab โ cd)x + b^2 + d^2
(E) None of the above Answer: (B) I solve these types of problems slightly differently than described in the Schaumโs guide. You can decide for yourself which way you prefer. First, convert 7x^2 โ 39 x โ 18 to x^2 โ 39 x โ 7 โ 18 = x^2 โ 39 x โ 126. Factor x^2 โ 39 x โ 126. Need to find c, d that satisfy cd = โ126 and c + d = โ39. Try -42 and 3. Now we know that x^2 โ 39 x โ 126 = (x + 3)(x โ 42). Since we multiplied by 7 in the first step, we need to divide by 7: (x +^37 )(x โ 427 )
Since 42 is divisible by 7, the second term becomes (x โ 6). However, 3 is not divisible by 7, thus we โslideโ the 7 in front of the x. Thus the first term becomes (7x + 3). Thus 7x^2 โ 39 x โ 18 can be factored as (7x + 3)(x โ 6). N.B.: This rule works only for quadratics. Question 13: Factor the following polynomial:
16 x^2 โ 49 y^2
(A) (x + 7)(16x + 7y^2 ) (B) (4x โ 7 y^2 ) (C) (4x โ 7 y)(4x โ 7 y) (D) (4x โ 7 y)(4x + 7y) (E) None of the above Answer: (D) Answering this question requires a simply application of the โdifferences-of-squareโ rule for factoring polynomials. Best to just memorize this formula as it generally useful. Any polynomial that is expressed as a โdifference-of-squaresโ (i.e., anything of the form a^2 โb^2 where a and b are general expressions), factors according to (a โ b)(a + b). In this case a = โ 16 x^2 = 4x and b = โ 49 y^2 = 7y. Applying the rules yields answer D: (4x โ 7 y)(4x + 7y).
Fractions
Question 14: Multiply the following rational expressions involving quotients of binomials and reduce to lowest terms. (^) x โ 5
x + 8 ยท^
x + 2 x โ 9 (A) ((xxโ+8)(5)(xx+2)โ9)
(B) x x^22 โโ^3 xxโโ 7210
(C) ((xxโ+8)(5)(xx+3)โ9)
(D) x x^22 +3โxx+72โ^10 (E) None of the above Answer: (B) Work through the following algebra:
x โ 5 x + 8.
x + 2 x โ 9 =
(x โ 5)(x + 2) (x + 8)(x โ 9) =^
x^2 โ 5 x + 2x โ 10 x^2 โ 9 x + 8x โ 72 =^
x^2 โ 3 x โ 10 x^2 โ x โ 72
Question 15: Divide the following expressions:
15 ch^4 5 x^2 yz^3 รท^
3 c^4 h 55 y^2 z
(A) (^) c^553 xh (^23) zy 2
(B) (^55) c 4 hxz^2 y 22
(C) (^55) c 3 hxz^3 y 2
(D) (^) c^553 xh (^22) zy 2 (E) None of the above Answer: (A) Work through the following algebra:
15 ch^4 5 x^2 yz^3 รท^
3 c^4 h 55 y^2 z =
515 x^2 chyz^43 553 cy^42 hz^ =^
15 ch^4 5 x^2 yz^3
55 y^2 z 3 c^4 h =
5 h^3 x^2 z^2
11 y c^3 =
55 h^3 y c^3 x^2 z^2
Question 16: Add or subtract the following fractions: 12 x^2 โ 49 +^
7 x x + 7 (A) (^) x 2 12+7+xโx 42
(B) 12((xx 2 +7)+7โ49)(xx+7)^3 โ^49
(C) 12(x+7)+7 x (^2) โx 49 (x^2 โ49)
(D) 12+7 x 2 x+49(xโ7) (E) None of the above Answer: (E) Work through the following algebra:
12 x^2 โ 49 +^
7 x x + 7 =^
(x โ 7)(x + 7) +^
7 x x + 7 =
12 + 7x(x โ 7) x^2 โ 49 =
12 + 7x^2 โ 49 x) x^2 โ 49
Question 17: Write the following expression in its simplest form:
x^2 โ 2 x + 1 x + 3 ยท^
x^2 + 5x + 6 x^2 โ 1
Answer:x^2 โ x+3^2 x+1 ยท x^2 x+5 (^2) โx 1 +6 = (x xโ+31) 2 ยท ((xx+3)(โ1)(xx+2)+1) = (xโ1)( x+1x+2)
Radicals
Question 18: Simplify the following the radicals: โ 81 x^8 y^6
(A) 9x^4 y^3 (B) 9โx^8 y^6 (C) โ 9 x^4 y^3 (D) 9x^4 y^3 and โ 9 x^4 y^3 (E) None of the above
