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/ 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 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 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 Example 1 To add the coefficients of like terms, and you can use a vertical or horizontal format

Example 2

Fk Adding Polynomials Video

Subtracting Polynomials Example 1 (2x2 - 4) - (x2 + 3x - 3) = (2x2 - 4) + (-x2 - 3x + 3) = 2x x2 - 3x + 3 = 2x2 - x2 - 3x = 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 g

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 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

Divide 2x 4 -9x 3 +21x x + 12 by 2x - 3 Example 1

Rational Zeroes Theorem We 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 x 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: =========================================== => Example 1

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