Polynomial Notes

Graphing

Solving

Writing

End Behavior

End Behavior

local maximum

local minimum

real root

x=10 or (x­10)

real root

x=­10 or (x+10)

real root

x=0 or (x+

0)

Long Division

­2

Rational Zeros

Factoring

Synthetic Division

Exponents

Polynomials Polynomials1. P­1: Exponents2. P­2: Factoring Polynomials3. P­3: End Behavior4. P­4: Fundamental Theorem of Algebra

Polynomial Notes

Properties of Exponents P1

Polynomial Notes

Adding and Subtracting PolynomialsAdding & Subtracting Polynomials is COMBINING LIKE TERMS. To be considered like terms, the terms must have the same variable, and the variables must have the same exponents. We add or subtract the coefficients, leaving the variables unchanged. All answers must be written in standard form, the largest exponent first, the rest in descending order.

Examples:Horizontal Format:

Vertical Format:

Polynomial Notes

Multiplication of PolynomialsDistributive PropertyMultiply both terms in the second parenthesis by the first term in the first parenthesis. Next, multiply each term in the second parenthesis by the second term in the first parenthesis. Combine like terms.

Vertical MethodWrite equations in standard form. Align like terms in columns. Multiply each term in the top equation by each term in the bottom equation. Combine like terms.

Polynomial Notes

Factoring Polynomials

P2

Algorithm:1. Split the problem into two parts.2. Factor the GCF out of each part.3. Factor the common parentheses out.4. Write as the product of two binomials.

Factor by Grouping

Polynomial Notes

Long DivisionAlgorithm:1. Write the function in standard form.

(Exponents in descending order, allow zeros as place holders)2. Make the leading term of the divisor exactly match the leading term of

the dividend using multiplication.3. Distribute the factor through the divisor and subtract.

(Subtract every term)4. Repeat steps 2 and 3 if necessary.5. Write any remainders in fraction form.

Example:

Check:

Polynomial Notes

­7

3

Multiply the outside numbers, write the answer in the next available spot inside the box

­21r

Synthetic SubstitutionAll equations must be written with decreasing exponents, biggest exponent first, constant last.

Rearrange if necessary

Make sure every exponent has a spot, Add a zero if necessary

=

=

Put coefficients in the box, bring the first one down

­7Put the number you want substituted in on the outside of the box

Add the inside numbers

­7

3­21­23

add

­7

3­21­23

­69

­69

­207

­202

­606

­597 ANSWER

Polynomial Notes

Synthetic DivisionAlgorithm: 1. Write the dividend in standard form. (Exponents in descending order.)

2. Solve the divisor for x.

3. Use synthetic substitution to determine if k is a zero of the polynomial function.

4. The result will be the coefficients of the quotient.

Examples:

Summary:

Polynomial Notes

Put this number in front

Synthetic Division

Summary:

­3 ­3

Solve the denominator for x

remainder

ANSWER

Put this number in front

­2 ­2

remaindercxx2x3

ANSWER

Solve the denominator for x

032

r.1

1

or

Polynomial Notes

Rational ZerosAlgorithm: 1. List all possible rational zeros.

The numerator represents all possible factors of the constant term.The denominator represents all possible factors of the leading coefficient.

2. Test all possible zeros using synthetic division. 3. Continue testing zeros until the result is a polynomial that can be factored by

grouping, or a trinomial that can be factored into two binomials. 4. If the resulting quadratic cannot be factored, use the quadratic formula to find the

remaining zeros. 5. List the real zeros of the function.

Example:

P4

Polynomial Notes

Summary:

Finding ZerosGiven one zero, find the others

;divide

factor

rewrite

solve

ANSWERS

Polynomial Notes

Real Zeros

Use your calculator to find one root

Find all real zeros of the function.

now use that to verify the other zeros (roots) using synthetic division

Polynomial Notes

Write the Equation of the Polynomial

Algorithm: 1. Write the given roots (intercepts) in factored form

x = 2 becomes (x ­ 2) 2. Use the distributive Property to multiply 3. Write the function in standard form* Remember all i 's must occur in pairs

Examples:

(Leading Coefficient is One)

Polynomial Notes

Write the Equation of the Polynomial

* Remember all i 's must occur in pairs

Examples:

(Leading Coefficient is NOT One)

Algorithm: 1. Write the given roots (intercepts) in factored form

x = 2 becomes (x ­ 2) 2. Substitute all roots, x and y into f(x), solve for a. 3. Use the distributive Property to multiply 4. Write the function in standard form

Polynomial Notes

Graphing Polynomial FunctionsAlgorithm:1. Find all rational zeros. Write the function in intercept form.2. Determine the end behavior of the function.3. Graph all rational zeros.4. Find all turning points

a. Turning points fall half way between two consecutive zeros. This will produce the x­value of the turning point.b. Substitute the x­value into the original function and solve for the y­value.

5. Plot the turning points.6. Draw a smooth curve through the points.

Example:

Polynomial Notes

Summary:

Graphing PolynomialsUse you calculator

Find the zeros

Find the local maximums and minimums

Sketch the graph

b 16

Zero must lie within shaded region

b62

Minimum must lie within shaded region

Plot the zerosPlot the local maximums or minimumsConnect the points