Download A. Simplifying Polynomial Expressions

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Algebra Skills Needed to Be Successful in College Algebra
A. Simplifying Polynomial Expressions — You need to be able to:
● Simplify an algebraic expression using the operations and properties
● Simplify polynomial expressions using addition and subtraction.
● Multiply a monomial and polynomial.
B. Solving Equations — You need to be able to:
● Solve multi-step equations.
● Solve a literal equation for a specific variable, and use formulas to solve problems.
C.
Rules of Exponents — You need to be able to:
● Simplify expressions using the laws of exponents.
● Evaluate powers that have zero or negative exponents.
D. Binomial Multiplication — You need to be able to:
● Multiply two binomials.
E. Factoring — you need to be able to:
● Identify the greatest common factor of the terms of a polynomial expression.
● Express a polynomial as a product of a monomial and a polynomial.
● Find all factors of the quadratic expression ax2 + bx + c by factoring and graphing.
F. Radicals — you need to be able to:
● Simplify radical expressions.
G. Graphing Lines — you need to be able to:
● Identify and calculate the slope of a line.
● Graph linear equations using a variety of methods.
● Determine the equation of a line.
H. Functions — you need to be able to:
● Work with functions in different forms — graphs, tables of values, words, and equations —
including reading points, interpreting the function in various ways, finding domain and
range, tc.
● Understand function notation and use it in different ways.
● Determine whether a point satisfies an equation and explain what that means for a graph.
A. Simplifying Polynomial Expressions
Combining Like Terms: You can add or subtract terms that are considered “like”, or terms that have the
same variable(s) with the same exponent(s).
Example 1:
5x – 7y + 10x + 3y = 15x – 4y
Example 2:
–8h2 + 10h3 – 12h2 – 15h3 = –20h2 – 5h3
Applying the Distributive Property: Every term inside the parenthesis is multiplied by the factor outside
of the parentheses.
Ex. 1: 3(9x – 4) = 3 • 9x – 3 • 4
Ex. 2: 4x2(5x3 + 6x) = 4x2 • 5x3 + 4x2 • 6x
= 27x – 12
= 20x5 + 24x3
Combining Like Terms AND the Distributive Property (Problems with a Mix!): Sometimes problems will
require you to distribute AND combine like terms.
Example:
3(4x – 2) + 13x = 3 • 4x – 3 • 2 + 13x
= 12x –6 + 13x
= 25x – 6
Practice Set A: Simplify each expression. Show your steps clearly using the method shown above.
1. 8x – 9y + 16x + 12y
2. 14y + 22 – 15y2 + 23y
3. 5n – (3 – 4n)
4. –2(11b – 3)
5. 10q(16x + 11)
6. –(5x – 6)
B. Solving Linear Equations
Solving Two-Step Equations:
To solve an equation, you can UNDO the order of
operations and work in the reverse order.
Addition is “undone” by subtraction, and vice
versa. Multiplication is “undone” by division, and
vice versa.
Simplifying Before Solving:
If variables are on both sides of the equal sign, get
all terms with variables on one side and all terms
without variables on the other side. You also may
need to use the distributive property.
5(4x – 7) = 8x + 45 + 2x
20x – 35 = 10x + 45
–10x
–10x
10x – 35 = 45
+ 35 +35
10x = 80
x=8
Ex. 1: 8x + 4 = 4x + 28
–4
–4
8x = 4x + 24
–4x –4x
h
4x = 24
x=6
Practice Set B1: Solve each equation. Show all steps like the examples above.
1. 5x – 2 = 33
2. 140 = 4x + 36
3. 8(3x – 4)=196
4. 45x – 720 + 15x = 60
5. 132 = 4(12x – 9)
6. 198 = 154 + 7x – 68
Solving Literal Equations: A literal equation is an equation that contains more than one variable. You
can solve a literal equation for one of the variables by isolating the specified variable (getting that
variable by
itself).
Practice Set B2: Solve each equation for the variable specified. Show all steps like the examples above.
7) Solve for a in the Pythagorean Theorem a2 + b2 = c2.
8) Solve for r in A = πr2
C. Rules of Exponents
Practice Set C: Simplify each expression.
D. Binomial Multiplication
When multiplying two binomials (an expression with two terms), you can distribute or use “FOIL”
method. (The “FOIL” method is actually the result of using the distributive property twice.)
= F + O + I + L = x2 + 10x + 6x + 60 = x2 + 16x + 60.
Practice Set D
Multiply. Write your answer in simplest form.
Explain why (x + a)2 is always equivalent to x2 + 2ax + a2.
E. Factoring
Using the Greatest Common Factor (GCF) to Factor
Applying the difference of squares: a2 – b2 = (a –b)(a + b
Practice Set E
F. Radicals
To simplify a radical, we need to find the greatest perfect square factor of the number under the radical
sign (the radicand) and then take the square root of that number.
Example:
If you don’t find the greatest square first, you may still be able to simplify the radical, but it will take
more steps, such as below.
Same Example:
Practice Set F: Simplify each radical.
G. Graphing Lines
Slope: There are many ways to represent slope. List as many ways as you can think of. You can use
words to describe what slope tells us about a line or a linear function. You can also use symbols and
formulas.
Slope
=
=
[use words]
=
[use other words]
=
[use symbols]
= the
= the
of the line [use a word]
of
of the y-value with respect to its x-value.
Use the graph patch to the right to sketch the points and
slope below. Then explain your list about slope using some
sentences and this example.
Practice Set G1: Find the slope between the given points.
Practice Set
G.2
III. Graphing a Line given in Standard Form — TWO OPTIONS:
Practice Set G3: Graph each line
H. Functions
1) This problem is about the function g(x) = –0.5x + 3.
y
a) Compute the following:
g(–4)
g(–2)
g(0)
g(0.5)
g(2)
g(4)
b) Make a list of the points (x, y) associated with
part (a) above. Your list can be separated by
commas.
c) Plot those points (x, y) on the grid to the right.
d) Do you think the pattern continues in between the points? Check and then change your
graph accordingly.
e) How would you express g(k)?
f) Challenge: Express g(–x2).
2) The equation x = 3 is NOT a function of x. Give as many reasons as possible.
x
3) Determine the values below using the graphs of f and g shown. If a value is not defined, state
“not defined.” Assume the ends of f and g continue the shape you see beyond the graph.
a) f(0)
b) g(0)
c) 2 – 5(g(0) –3) (Read carefully.)
d) g(3)
e) g(–2)
f) f(1) + g(1)
g) f(–3) + g(5)
h) For what values of x does g(x) = –2?
i) For what values of x is f decreasing?
j) For what values of x is g increasing?
k) Give three values that g(x) can never be? (Is this asking for an x-value or a y-value?)
l) Describe all the possible output values of f in a sentence.
4) What does the notation f(x) really represent? Is it a function? An x-value? A y-value? An
expression? (for what?) A complete equation? (for what)? Explain.
5) Use the graph below of y = f(x) to answer
the following questions.
a)
What does the closed point at the left end of
the graph tell you?
b) What does the arrow tell you?
c)
What is the domain of f?
d) What is the range of f?
e)
For what value(s) of x does y = 0?
f)
An x-intercept is an x-value where the graph crosses the x-axis. Identify the x-intercepts of f.
g)
What is the y-intercept?
h) A function can have multiple x-intercepts but only one y-intercept? Explain why using the
definitions of the word function, x-intercept, and y-intercept.
i)
For what values of x is f(x) < 0?
j)
Graph g(x) = –3. (Reason this out — this equation states that no matter what x is, g(x) is…?)
k) For what values of x is g(x) ≥ f(x)?
l)
Explain why it is impossible to write up a complete table of values for this graph — and why
that’s not really a problem.
6) For any function f, Does the point (x, f(x)) satisfy the equation that represents f(x)?
7) Which of the following equations does the point (x, y) = (5, 10) satisfy?
a) f(x) = 10x – 50
b) g(x) = 2x
c) h(x) = x2 – 15
d) j(x) = 5
e) k(x) = 10
8) Is the point (–9, 3) on the graph of
9) Is the point (4, 3) on the graph of
? Explain.
? Explain.
10) What does it mean for a point to “satisfy” an equation?
Answer Key
Practice Set A
Practice Set B1
Practice Set B2
Practice Set C
Practice Set D
Practice Set E
Practice Set F
1. 11
2.
3.
4.
5. 8
6.
7.
8.
9.
Practice Set G1
Practice Set G2
Practice Set G3