Download Math 31 Answers to Test 3 (I)

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