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A collection of quiz questions from the SGPE Summer School 2016 covering topics such as exponents, polynomials, and radicals. Students are required to simplify expressions using various rules for manipulating exponents, expand and regroup expressions, factor polynomials using integer coefficients, and perform arithmetic operations on fractions.
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Typology: Lecture notes
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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
Answer:
(xโ (^14) y 3 x 12
= (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