(Solved by Humans)-Algebra help . Need work shown and answers circled.

Question

Algebra help . Need work shown and answers circled.?

Stony Brook University
Mathematics Department
Jerson R. Cochancela, Julia Viro

Proficiency Algebra
MAT 103, Spring 2016

Practice Final Exam
Part I
1. For each of the following expressions state if it is a polynomial or not. For polynomials,
indicate their degrees.
a) 7 6x5 ? x4 + 2x3 ? 4x + 5

b) ?x3

c) ?11x5 + 6x4 + 2x?3 ? 1

?
2x3 ? 4x2 + 10
e) 6x5 ? x4 + 2 x + 5
e
2. Simplify the following expressions:
?
?
?
?
?
?
3 75
a)
? 5 27
b) 3 64 ? 250
c) (1 ? 7)(1 + 7)
2
?
?
3
e) (?3 2x)2
f ) (? 8x2 )3
d)

d)

?

?
?
3(2 3 + 6)

3. Express in simplest radical form, rationalize if necessary.
?
?
1
9 2
2+ 3
?
?
a)
b) ?
c)
1+ 2
3
2? 3
4. Solve the following quadratic equations:
a) x2 ? 16 = 0

x
c) x2 ? 8 = 0 d) ?3x2 + x = 0 e) x2 ? = 0
4
?
g) ?x2 ? 1 = 0 h) 2 x2 + ?x = 0

b) x2 + 16 = 0

f ) 2014 x2 + 2015 = 0

5. Compose a quadratic equation whose roots are
?
?
a) 2 and ?4
b) ?5 (a double root)
c) 5 and ? 5.
6. Factor the following quadratic: y = 2x2 + 12x + 13 (hint: solve for roots using the quadratic
formula, then use roots to find factors)
7. For the parabolas defined by the following equations, find the vertex, the axis of symmetry,
the intercepts, and draw the graphs.
1
d) y = ? x2
3
8. Let p(x) = x2 ? 3x + 1 and q(x) = x3 + 2 be polynomials. Determine the degrees of p(x)
and q(x). Find

a) y = x2 + 2x ? 3 b) y = 2x2 ? 2x + 4

a) p(x) + q(x)

b) p(?1) + 3q(2)

c) y = x2 ? 4x + 4

c) 2p(x) ? q(?1)

d) p(x)q(x)

e)

p(x + 1)
2q(0)

2

Part II
1. Expand and combine similar terms:
a) (x2 ? 3x + 2)(?x + 1)

b) (xy 2 + 1)(x2 y + x ? 1)

c) (x2 + xy + y 2 )(x ? y)

2. The Lazy Snail, a shipment company, calculates the total cost of shipments per day by the
formula y = 41 x2 ? 2x + 10, where x is the number of shipped packages and y is the cost in
dollars. Find the number of packages minimizing the cost of shipments per day. What is this
minimal cost?
3. In a right triangle, one leg measures 4 cm. The hypotenuse is 2 cm shorter than the other
leg. Find the perimeter and area of the triangle (hint: use the Pythagorean theorem)
4. The parabola y = ax2 + bx + c has no x-intercepts. Its vertex is located in the second
quadrant. Determine the signs of a, b and c.
5. One side of a rectangle is 5 cm shorter than the other side. Find the perimeter of the
rectangle if its area in 24 cm2 .
6. Solve the following systems. Check your solution (if any) by a substitution. Give a geometric
interpretation of the system and its solution.
(
(
2x + 3y = ?8
2x + 1 = y
?3x + 4y = ?39
y + 2 = x2
7. Simplify the following expressions:
x3 ? x
a) 5
x ? x3

x2 + 18xy + 81y 2 1
b)
3x2 + 27yx

3x ? 15 10x ? 20x2
c) 2
?
4x ? 2x
5?x

8. Solve the following inequalities:
a) x2 ? 3

b) x2 > 3

c) 3x2 ? 2x ? 0

d) 3x2 ? 2x < 0