Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math 31 (1) Multiplying: Answer: Answers to Test 3 (I) (4x + 3)(16x2 − 12x + 9) (D) 64x3 + 27 16x2 −12x × 4x +3 Work: +9 64x3 −48x2 +36x + 48x2 −36x +27 64x3 (2) −x2 y 3 − x3 y 2 − 21 x4 Evaluating: Answer: (A) +27 for x = −2 and y = −1 4 Work: −x2 y 3 − x3 y 2 − 12 x4 = −(−2)2 (−1)3 − (−2)3 (−1)2 − 12 (−2)4 = −4(−1) − (−8) · 1 − 21 · 16 =4+8−8=4 (3) Expanding (multiplying out): Answer: Work: (4) (B) 16x4 − 48x3 + 36x2 (4x2 − 6x)2 = (4x2 )2 − 2(4x2 )(6x) + (6x)2 = 16x4 − 48x3 + 36x3 Subtracting: Answer: (4x2 − 6x)2 (10x3 − 8x2 + 7x − 12) − (13x3 − 5x2 + 9x − 3) (C) −3x3 − 3x2 − 2x − 9 Work: (10x3 − 8x2 + 7x − 12) − (13x3 − 5x2 + 9x − 3) = (10x3 − 8x2 + 7x − 12) + (−13x3 + 5x2 − 9x + 3) = 10x3 − 8x2 + 7x − 12 − 13x3 + 5x2 − 9x + 3 = −3x3 − 3x2 − 2x − 9 (5) What is the degree of the polynomial: Answer: (D) 15x2 y 7 − 9x5 y 3 9 Note: The degree of a monomial is the exponent of the variable (or the sum of exponents of the variables.) The degree of a polynomial is the highest degree of the monomials in that polynomial. (6) Simplifying: Answer: (5a2 )2 (−3a−4 )3 (C) − 675 a8 Work: (5a2 )2 (−3a−4 )3 = 52 a4 (−3)3 a−12 = 25 · (−27)a4+(−12) = −675a−8 (7) Write the number 0.0000000734 in scientific notation. Answer: (8) (A) 7.34 × 10−8 Rewrite the number 0.0028 × 109 in scientific notation. Answer: Work: (D) 2.8 × 106 (rewrite 0.0028 in scientific notation first) =⇒ 0.0028 × 109 = 2.8 × 10−3 × 109 = 2.8 × 10−3+9 = 2.8 × 106 (9) Simplifying: Answer: 2−2 − 2−1 (B) −2−2 Work: 2−2 − 2−1 = 212 − 12 = 14 − 21 = 14 − 24 = − 14 = −2−2 (10) Factoring: x2 + 7x − 30 (11) Factoring: 16x2 − 25 Answer: (A) Answer: (C) (x + 10)(x − 3) (4x + 5)(4x − 5) Work: By the formula a2 − b2 = (a + b)(a − b) =⇒ 16x2 − 25 = (4x)2 − (5)2 = (4x + 5)(4x − 5) (12) Factoring: x2 + 7x + 1 (13) Factoring: 8x2 y 2 − 18y 4 Answer: (D) Answer: Prime (It can not be factored.) 2y 2 (2x − 3y)(2x + 3y) (B) Work: 8x2 y 2 − 18y 4 = 2y 2 (4x2 − 9y 2 ) = 2y 2 ((2x)2 − (3y)2 ) = 2y 2 (2x − 3y)(2x + 3y) (14) 8x3 + 64 Factoring: Work: Answer: 8(x + 2)(x2 − 2x + 4) (D) According to the formula a3 + b3 = (a + b)(a2 − ab + b2 ) 8x3 + 64 = 8(x3 + 8) = 8(x3 + 23 ) = 8(x + 2)(x2 − 2x + 22 ) = 8(x + 2)(x2 − 2x + 4) (15) Solving the quadratic equation: Work: 25x2 = 64 Answer: { 58 , − 58 } (A) 25x2 = 64 =⇒ 25x2 − 64 = 0 =⇒ (5x + 8)(5x − 8) = 0 =⇒ Either 5x + 8 = 0 or 5x − 8 = 0 or x = ( or x112 ) (17) x3 y −5 x−4 (y 2 )4 (18) 1 1 −1 1 1 (−30 )(−3)−1 (−3)−2 = (−1)· −3 · (−3) 2 = (−1) ( 3 )( 9 ) = 27 (19) −50 − 5−1 + (−5)−2 = x3 y −5 x−4 y 8 = x3−(−4) y−5−8 = x7 y−13 1 = −1 − 51 + 25 = −25 25 = x−12 8 5 x2 · x · (x−5 )3 = x2+1+(−15) −8 5 (16) = x2 · x1 · x−15 =⇒ x = 7 ( or yx13 ) − 5 25 + 1 25 = −25−5+1 25 = − 29 25 (20) 3(2x2 −3x+7)−2(8x2 −5x+2) = (6x2 −9x+21)−(16x2 −10x+4) = 6x2 −9x+21−16x2 +10x−4 = −10x2 + x + 17 (21) (5x − 3y)(3x + 5y) = 15x2 + 25x − 9x − 15y 2 = 15x2 + 16xy − 15y 2 (22) (2x2 − 5x)2 = (2x2 )2 − 2(2x2 )(5x) + (5x)2 = 4x4 − 20x3 + 25x2 (23) (3x2 − 5x + 2)(7x2 + 3x − 4) = 21x2 − 26x3 − 13x2 + 26x − 8 (24) Divide: 6x3 −5x2 +10x−7 3x−1 Work: 3x − 1 ) −4 = (2x2 − x + 3)+ 3x−1 2x2 −x 3 2 +3 6x −5x +10x −7 6x3 −2x2 −3x2 +10x −3x2 +x 9x −7 9x −3 −4 (FOIL method) (25) Factoring: 3x2 − 7x − 6 ac-method: Step 1: Step 2: Step 3: (26) a = 3 and c = −6 =⇒ ac = −18 Find two numbers p and q such that pq = −18 and p + q = −7 3x2 − 7x − 6 = 3x2 − 9x + 2x − 6 = (3x2 − 9x) + (2x − 6) = 3x(x − 3) − 2(x − 3) = (x − 3)(3x + 2) Factoring: =⇒ p = 2 and q = −9 15x5 − 9x4 + 30x3 − 18x2 15x5 − 9x4 + 30x3 − 18x2 = 3x2 (5x3 − 3x2 + 10x − 6) = 3x2 ((5x3 − 3x2 ) + (10x − 6)) = 3x2 (x2 (5x − 3) + 2(5x − 3)) = 3x2 (5x − 3)(x2 + 2) (27) Solving the quadratic equation: x2 − 10x − 24 = 0 Factor the left hand side of x2 − 10x − 24 = 0 =⇒ (x − 12)(x + 2) = 0 =⇒ x = 12 or x = −2