1. Try Factoring – sometimes it might be possible.
2. Type the function into your Y = screen on the calculator and view the graph to
find one zero.
3. Double check this zero by looking at your table. That x-value should have a 0
in the y-column.
4. Do synthetic division with that zero.
5. Write a new polynomial with remaining coefficients.
6. Factor or use the Quadratic Formula with remaining polynomial to find the rest
of the zeros.
Rational Zero Theorem (5.8)
1. Determine all of the solutions of the function. (Don’t forget that imaginary roots
always come in pairs)
2. Write the solutions as factors.
3. Multiply all factors by using either the Box Method or Double Distribute.
4. Write your final answer as f(x) = …
Writing a Polynomial Function (5.7)
Property
Definition
Product of Powers
xa  xb = xa + b
Quotient of Powers
xa
 x a b , x  0
b
x
Negative Exponent
xa 
1
1
and  a  x a , x  0
a
x
x
Power of a Power
(xa)b = xa •b
Power of a Product
(xy)a = xaya
a
Power of a Quotient
Zero Power
x
xa

, y  0, and
 
ya
 y
x
 
 y
a
a
ya
 y
   or a , x  0, y  0
x
x
x0 = 1, x ≠ 0
Properties of Exponents (5.1)
Example
1. Rewrite the problem.
2. Divide the first term of the dividend by the first term of the divisor and
write that answer on the line.
3. Multiply that answer by the first term of the divisor and write your
answer below the dividend. Also multiply that answer by the second term
of the divisor and write your answer.
4. Subtract and bring down the next term from the dividend.
5. Divide the first term of this new dividend by the first term in the divisor
and write that answer on the line.
6. Multiply that answer by the first term of the divisor and write your
answer below the dividend. Also multiply that answer by the second term
of the divisor and write your answer.
7. Subract.
8. Your answer is on the line (at the top of the problem). If you have a
remainder, write it as a fraction at the end of your answer. The
denominator of that fraction is the divisor.
Dividing Polynomials using Long Division (5.2)
Degrees of polynomials
Rules to be a polynomial






Constant – Degree 0
Linear – Degree 1
Quadratic – Degree 2
Cubic – Degree 3
Quartic – Degree 4
Degree n – higher degree

Adding Polynomials
No:
1.
2.
3.
square roots of variables
fractional exponents
variables in the denominator of any
fraction
Subtracting Polynomials
1. Distribute the negative
to the second polynomial.
1. Combine like terms.
2. Combine like terms.
Distributing
Multiplying Polynomials
1. Determine whether you will need to distribute (one term times a polynomial)
OR
Box method (two or more terms times a polynomial).
*If multiplying a binomial by a binomial, make a 2x2 box, if a trinomial times
a binomial, make a 3x2 box
2. Combine like terms if needed.
Add/Subtract/Multiply Polynomials (5.1)
Box Method
A Relative Maximum is a point on the graph of a
function where no other nearby points have a greater ycoordinate.
A Relative Minimum is a point on the graph of a function
where no other nearby points have a lesser y-coordinate.
Extrema are the maximum and minimum values of a
function. These points are usually referred to as turning
points. It is also another name for relative maximums and
minimums. The graph of a polynomial function of
degree n has at most n – 1 turning points.
Find Extrema on Calculator:
1. Enter the equation into y =.
2. Hit 2nd Calc.
3. Choose 3: minimum or 4: maximum.
4. Put the curser to the left of min/max, hit enter.
5. Put the curser to the right of min/max, hit enter.
6. Hit enter.
Analyze the graph
Find the relative extrema of:
Relative Minimum:
Relative Maximum:
Describe the end behavior
Analyzing Graphs (5.3 & 5.4)
Equations of higher degree where the first term exponent is double the second term exponent
Decide what u represents (hint: Look at the second term
exponent)
Rewrite the equation as: au2 + bu + c = 0
Factor the equation using product sum or slide and divide
Substitute the x’s back in for the u.
Quadratic Form (5.5)
The binomial x – r is a factor of the polynomial P(x) if an d only if P(r) = 0
Used to factor polynomials with higher degrees that cannot be written in quadratic form
1. Perform synthetic division with the given factor.
2. Factor the remaining polynomial using product sum or slide
and divide
The Factor Theorem (5.6)
1. Write the coefficients of the dividend so that the degrees of
the terms are in descending order. Write the constant r of the
divisor x – r in the box.
Example: Divide x4 + 3x3 –x2 + 8 by x – 1
2. Bring the first coefficient down and multiply it by r. Write
the product under the second coefficient.
3. Add the product and the second coefficient.
4. Repeat Steps 2 and 3 until you reach a sum in the last
column.
5. The numbers along the bottom row are the coefficients of
the quotient. The power of the first term is one less than the
degree of the dividend. The final number is the remainder.
Answer:
Synthetic Division (5.2)
Sum of two cubes
a3 + b3 = (a + b)(a2 – ab + b2)
1. Factor out the GCF
2. Write each term as a perfect cube
3. Identify the given variables, a, b, a2, b2, ab
4. Substitute the values from step 3 into the formula
Difference of two cubes
a3 – b3 = (a – b)(a2 + ab + b2)
1. Factor out the GCF
2. Write each term as a perfect cube
3. Identify the given variables, a, b, a2, b2, ab
4. Substitute the values from step 3 into the formula
Sum & Difference of Cubes (5.5)
1. Write the expression in degree descending order.
2. Factor out a GCF, if necessary.
3. Group the first two terms together in parentheses and the second two terms
together in parentheses.
4. Factor out the GCF from each group.
5. Once the GCF is factored out, the items in the parentheses should be identical.
Write that here →
6. What’s left over should be written here and your answer from step 5 should be
written next to it.
Factoring by Grouping (5.5)