PolynomialsBy 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 substitutionTo solve a system via substitution:
1) Take one of the equations, and solve it for one variable in terms
of the other2) Plug this into the second
equation3) Solve this for the second
variable4) Now that you know one of the
variable values, plug it into either equation and solve for the other.
Example 13x-4y=0
3x+2y=28
Change 3x - 4y = 0 to 3x = 4y and x = (4/3)y
Plugging this into "3x + 2y = 28" gives you 3(4/3)y + 2y = 28
Solve this for y to get4y + 2y = 28
6y = 28y = 28/6Y = 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 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 BehaviorPolynomial 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
Example 1
To add the coefficients of like terms, and you can use a vertical or horizontal format
Example 2
http://www.youtube.com/watch?v=nhpXTQlwvFk
Adding Polynomials
Video
Subtracting Polynomials
Example 1(2x2 - 4) - (x2 + 3x - 3)
= (2x2 - 4) + (-x2 - 3x + 3)
= 2x2 - 4 + -x2 - 3x + 3
= 2x2 - x2 - 3x - 4 + 3
= x2 - 3x – 1
Change signs of terms being
subtracted and change subtraction
to addition.
Identify like terms
Group the like terms
Add the like terms
Change the signs of ALL of the terms being subtracted. Change the subtraction sign to addition. Follow the rules for adding signed numbers.
To subtract the coefficients of like terms, and you can use a vertical or horizontal method
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
Example 2
Subtracting Polynomials
Video
http://www.youtube.com/watch?v=fnCv6kWw4Eg
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 groupingra + rb + sa + sb= r(a + b) + s(a +
b) =( r + s)(a + b)
Polynomial Long Division
Divide the highest degree term of the polynomial by the highest degree term of the
binomial.
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.
Repeat this process until the remaining polynomial has lower degree than the binomial.
Polynomial Long Division Video
http://www.youtube.com/watch?v=FTRDPB1wR5Y
Divide 2x 4 -9x 3 +21x 2 - 26x + 12 by 2x - 3
Example 1
Rational Zeroes TheoremWe can use the Rational Zeros Theorem to find all the rational zeros
of a polynomial. Here are the steps:
Arrange the polynomial in descending order
Write down all the factors of the constant term. These are all the possible values of p .
Write down all the factors of the leading coefficient. These are all the possible values of q .
Write down all the possible values of . Remember that since factors can be negative, and - must both be included.
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) .
Steps
Find all the rational zeros of P(x) = x 3 -9x + 9 + 2x 4 -19x 2 .
P(x) = 2x 4 + x 3 -19x 2 - 9x + 9
Factors of constant term: ±1 , ±3 , ±9 .
Factors of leading coefficient: ±1 , ±2 .
Possible values of : ± , ± , ± , ± , ± , ± . These can be simplified to: ±1 , ± , ±3 , ± ,
±9 , ± .
Use synthetic division:
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Example 1
Sources
• http://www.mathsisfun.com/algebra/polynomials.html
• http://www.purplemath.com/modules/polydefs.htm
• http://www.purplemath.com/modules/polymult