Transcript Polynomials Overview

Polynomials
By Nam Nguyen,
Corey French, and
Arefin
Definition of a
Polynomial
A polynomial is an expression of
finite length constructed from
variables and constants,using only
the operations of addition,
subtraction, multiplication, and
non-negative integer exponents.
Example of a Polynomial
2x^2 − x/4 + 7
2 is the coefficient
 7 is the constant term
^2 is the degree

Direct substitution
To solve a system via substitution:
1) Take one of the equations, and solve
it for one variable in terms of the
other
2) Plug this into the second equation
3) Solve this for the second variable
4) Now that you know one of the
variable values, plug it into either
equation and solve for the other.
Example 1
3x-4y=0
3x+2y=28
Change 3x - 4y = 0 to 3x = 4y and x = (4/3)y

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Plugging this into "3x + 2y = 28" gives you 3(4/3)y + 2y = 28
Solve this for y to get
4y + 2y = 28
6y = 28
y = 28/6
Y = 14/3

Now you just have to find x. We said x = (4/3)y, so that means
x = (4/3)(14/3) = 56/9

Synthetic substitution

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Write the polynomial in descending order, adding "zero terms" if an exponent term is
skipped.
If the polynomial does not have a leading coefficient of 1, write the binomial as b(x - a)
and divide the polynomial by b . Otherwise, leave the binomial as x - a
Write the value of a , and write all the coefficients of the polynomial in a horizontal line
to the left of a


Draw a line below the coefficients, leaving room above the line.
Bring the first coefficient below the line.
Multiply the number below the line by a and write the result above the line below the
next coefficient


Subtract the result from the coefficient above it.
Repeat steps 6 and 7 until all the coefficients have been used.
If the polynomial has n terms, the first n - 1 numbers below the line are the
coefficients of the resulting polynomial, and the last number is the remainder.

Example 1
What is the result when 4x^4 -6x^3
-12x^2 - 10x + 2 is divided by x - 3
? What is the remainder?
End Behavior
Polynomial End Behavior
If the degree n of a polynomial is even,

then the arms of the graph are either
both up or both down
If the degree n is odd, then one arm of

the graph is up and one is down
If the leading coefficient an is positive,

the right arm of the graph is up
If the leading coefficient an is negative,

the right arm of the graph is down
Adding Polynomials
To add the coefficients of like terms,
and you can use a vertical or horizontal
format
Example 1
Example 2
Adding Polynomials
Video
http://www.youtube.com/watch?v=nhpXTQlwv
Fk
Subtracting Polynomials
To subtract the coefficients of like terms, and you can use a vertical or horizontal
method
Example 1
Change the signs of ALL of the terms being
subtracted. Change the subtraction sign to addition.
Follow the rules for adding signed numbers.
(2x2 - 4) - (x2 + 3x - 3)
= (2x2 - 4) + (-x2 - 3x + 3)
Change signs of terms being

subtracted and change subtraction to
addition.
= 2x2 - 4 + -x2 - 3x + 3
Identify like terms

= 2x2 - x2 - 3x - 4 + 3
Group the like terms

= x2 - 3x – 1
Add the like terms

Example 2
Using the vertical method to subtract like terms:
2x² + 0x - 4
-(x²+ 3x - 3)
Now, change signs of all terms being
subtracted and follow rules for add.
2x² + 0x - 4
-x² - 3x + 3 (signs
changed)
= x² - 3x - 1
Subtracting
Polynomials Video
http://www.youtube.com/watch?v=fnCv6kWw4E
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Special Product Patterns
= Sum and Difference of a binomial
(a + b)(a - b)= a^2 – b^2
=Square of a binomial
(a + b)^2= a^2 + 2ab + b^2
(a - b)^2= a^2 - 2ab + b^2
=Cube of a binomial
(a + b)^3= a^3 + 3a^2b + 3ab^2 + b^3
(a -b)^3= a^3 -3a^2b + 3ab^2 - b^3
Special Factoring
Patterns
= Sum and Difference of two cubes
a^3 + b^3= (a + b)(a^2 - ab + b^2)
a^3 - b^3= (a - b)(a^2 + ab + b^2)
= Factor by grouping
ra + rb + sa + sb= r(a + b) + s(a + b)
=( r + s)(a + b)
Polynomial Long Division
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Divide the highest degree term of the polynomial by
the highest degree term of the binomial.
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Write the result above the division line.
Multiply this result by the divisor, and subtract the
resulting binomial from the polynomial.
Divide the highest degree term of the remaining
polynomial by the highest degree term of the binomial.

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Repeat this process until the remaining polynomial has
lower degree than the binomial.
Polynomial Long
Division Video
http://www.youtube.com/watch?v=FTRDPB1wR5Y
Example 1
Divide 2x 4 -9x 3 +21x 2 - 26x + 12 by 2x - 3
Rational Zeroes Theorem
We can use the Rational Zeros Theorem to find all the rational zeros of a
polynomial. Here are the steps:
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Arrange the polynomial in descending order
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Write down all the factors of the constant term. These are all the possible
values of p .
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Write down all the factors of the leading coefficient. These are all the
possible values of q .
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Write down all the possible values of . Remember that since factors can be
negative, and - must both be included.
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Simplify each value and cross out any duplicates.
Use synthetic division to determine the values of for which P() = 0 . These
are all the rational roots of P(x) .
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Example 1
Steps
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Find all the rational zeros of P(x) = x 3 -9x + 9 +
2x 4 -19x 2 .
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P(x) = 2x 4 + x 3 -19x 2 - 9x + 9
Factors of constant term: ±1 , ±3 , ±9 .
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Factors of leading coefficient: ±1 , ±2 .
Possible values of : ± , ± , ± , ± , ± , ± .
These can be simplified to: ±1 , ± , ±3 , ± , ±9
, ± .
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Use synthetic division:
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Sources
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http://www.mathsisfun.com/algebra/polynomials.html
http://www.purplemath.com/modules/polydefs.htm
http://www.purplemath.com/modules/polymult