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