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2
Semester Review
Solve Equations by Factoring
Factoring Techniques
GCF
10y⁵ + 15y³
Difference of Squares
x² – 36
Trinomial
x² – 3x – 10
Quadratic Formula
How to solve Quadratic Equation.
1) Graph
Difficult to do without a graphing calculator, real numbers only.
2) Factoring
Only works if the trinomial/binomial is factorable, real
numbers.
3) Square root
Only works for x² equations without a middle term, real or
imaginary numbers.
4) Quadratic Formula
Can be used for any equation in the form ax² + bx + c = 0.
Real or imaginary numbers.
Solving quadratic equations
1) x² - 16 = 0
2) x² - 6x + 2 = 0
3) x² - 16x + 64 = 0
4) x² - 13 = 0
5) 4x² – x = 0
6) 2x² – 7x + 1 =0
Properties of Square Roots
How to simplify square roots.
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–
–
–
Rewrite the radicand into prime factors.
Pairs of prime factors will be moved out of the radical.
Pairs of variables will be moved out of the radical also.
All non-pairs are left under the square root.
Square Roots of Negative Numbers
Imaginary Unit (i)- negative numbers under the square root.
i = √ (-1)
i² =
i³ =
i⁴ =
Graph Quadratic Function
Quadratic Function
Is an equation written in the form
f(x) = ax² + bx + c where a ≠ 0
Graph - is a parabola.
Y-intercept- where the graph crosses the y-axis
Axis of Symmetry- imaginary line where the parabola
folds onto its self.
Vertex-the point where the axis of symmetry and the
parabola intersect.
Domain:
Range:
How to Graph
f(x) = x² + 8x + 9
Y-intercept (0,c)
Axis of Symmetry
Table of Values
• Include the x-coordinate of the vertex
• Two values on each side of the vertex
Properties of Powers
Negative
Product of Powers
Quotient of Power s
Power of Power
Zero Power
Polynomial
What is a polynomial?
No dividing of variables.
No negative exponents.
No variables under a radical symbol.
Degree of a polynomial
1. Find the degree of each monomial(term)
 Add the exponents on all the variables in the monomial
2. The largest degree is also the degree of the polynomial
Simplify Polynomials
Add/subtract polynomials
Combine like terms
• Coefficients change
• Variables stay the same.
Be careful of subtraction signs.
1) (2a³ + 5a – 7) – (a³ – 3a + 2)
Simplifying Polynomials
Multiply Polynomials
1. Distributive property or FOIL
• Use the rules for multiplying monomials
• Multiply Coefficients
• Add exponents
2. Combine like term
1) 2p²q(5pq – 3p³q² + 4pq⁶)
2) (y – 10)(y + 7)
3) (m – 3)²
Graphs of Polynomial Functions
Constant
Linear
Quadratic
Cubic
Quartic
Quintic
Examples
For each graph
a. Describe the end behavior
b. Determine whether it represents an odd or even degree function.
c. State the number of real zeros
GCF and Difference of Squares
ALWAYS look for a GCF before trying other
factoring techniques. It will usually make any
future factoring easier.
Difference of Squares a² - b² = ( + )( - )
• Must be subtraction
• Even exponent on variables
• Numbers are perfect squares
Perfect Cubes
Factoring perfect cubes
•
•
•
•
Variable has an exponent of 3
All numbers are a perfect cubes
Sum of Two cubes
a³ + b³ = (a + b)(a² – ab + b²)
Difference of Two cubes
a³ – b³ = (a – b)(a² + ab + b²)
Trinomial Factoring
Trinomial Factoring
•
•
•
•
ax² + bx + c = (
ax² + bx - c = (
)(
)(
Two parentheses
Factors of the first term
Factors of the last term
Combination of factors (O & I) use equal the middle
term and determine the signs.
)
)
Factoring by Grouping
Factoring by grouping is a factoring method that can be
used with four term polynomials.
• Regroup terms 1 & 2 together and 3 & 4 together.
• Find a GCF for each group
• If parentheses match you can use factoring by grouping. Write the
multiplication problem as (matching parentheses)(outside parentheses)
1) x³ + 5x² + 2x + 10
2) x² + 3xy + 2xy² + 6y³
Factoring Techniques
• Whenever you factor a polynomial, always look for a
GCF first! Then determine whether the resulting factor
can be factored again using a different method.
• Determine the appropriate factoring method based on
the number of terms and exponents.
2 terms
3 term
4 terms
• If none of the methods the factoring techniques will
apply then the polynomial is called prime.
Division
Polynomial ÷ Monomial
Rewrite the division problem as individual monomial division
problems and simplify.
1) 9x²y³ – 15xy² + 12xy³
3xy²
2) 16a⁵b³ – 20ab⁵
4ab⁷
Synthetic Division
Polynomial ÷ Polynomial
• When dividing polynomials they must be
written in descending order and every
degree must be accounted for.
1. (x² + 7x – 30) ÷ (x – 3) 2. (3a⁴ – 6a³ – 2a² + a – 6) ÷ (a + 1)
Composition of Functions
[f g](x) = f[g(x)]
Note g[f(x)] is usually a different composition from above.
Start with the inside function first and work your way out.
Evaluate f[g(x)] and g[f(x)]
1. f(x) = 2x
g(x) = 3x – 4
Given g(x) = -3x and f(x) = x² + 2x, find
4) f[g(2)]
5) g[f(4)]
Inverse Relations
Two relations are inverse relations if and only if whenever
one relation contains the element (a,b), the other
relation contains the element (b,a).
Q = {(1,2), (3,4), (5,6)}
How to Find an inverse of a function
1) Replace f(x) with y in
the original equation.
2) Interchange x and y.
3) Solve for y.
f(x) = -½x + 1
Square Root Functions
• Contains a square root of a variable.
Domain: All real numbers ≥ x-value of the end point.
Range:
Graph goes up: All real numbers ≥ y-value of the endpoint.
Graph goes down: All real numbers ≤ y-value of the endpoint.
How to Sketch a graph
1. Determine where the radicand = 0.
This is the x-value of end point of the graph.
2) Make a table of values that starts with the value found
in step 1 and numbers that are greater than it.
3) Plot the points found in the table.