polynomials - university place school district / overview€¦ · · 2016-12-06polynomials...
TRANSCRIPT
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
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: