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Chapter 5 : Polynomials and Polynomial Functions
5.1 Properties of Exponents
Rules:
1. Product of Powers: Add the exponents, base stays the same
2. Power of Power: Multiply exponents, bases stay the same
3. Power of a Product: Separate letters, distribute to the exponents and the bases
4. Zero exponents: Any number to the zero power = 1, zero to any power is o
5. Negative exponents: If base has (), change to reciprocal, no () in final answer
6. Quotient of Powers: Subtract exponents, simplify bases if possible
7. Power of a Quotient: Find the power of the numerator and denominator
Hints:
● If a base doesn’t have an exponent, it is to the 1st power● Follow order of operations (PEMDAS)● Reciprocals as last step
● Multiply straight across● Reduce everything to lowest terms● The answer should have only one of each base remaining● No negative exponents in answer
To Simplify Expressions:
● Use the properties to get rid of ( ).● Multiply fractions straight across, line up common bases.● Reduce the numbers, move negative exponents, get to one of each base.
Scientific Notation
• Negative powers,start small
Ex. 1.25 * 10¯³ =.00125
• Positive powers,start big
Ex. 1.25 * 10³ = 1250
Exponential Growth:
A = C( 1 + r)t
Exponential Decay:
A = C(1 r)t
5.2 Evaluate and Graph Polynomial Functions
5. 3 Add, subtract and multiply polynomials
● Monomial: a single term with exponent as integer● Polynomial: Has many(all) terms with exponents as integers● Coefficients: numbers that are in front of terms● Leading Coefficient: coefficient of term with highest exponent (+) opens up, () opens
down● Terms: Variables separated by addition term the amount of terms determines what type
(monomial, binomial, trinomial, etc)● Degree: Value of the highest exponent (Linear is single term) 3 = cubic, 2 = quadratic 1 =
monomial● Constant: the single number at the end (no letters attached to it)
● Adding : Combine like terms
(2x + 3) + ( 3x + 4) = 5x + 7
● Subtracting: Change subtraction to adding the opposite
( 2x + 3) – ( 3x + 4) → ( 2x +3) + ( 3x – 4) = 1x – 1
● Multiply: Use foil, combine like terms – First, outside, Inside, last or distribute each termsinglely. With multiple terms, just distribute and combine like terms.
(2x+3)(3x+4) =
6x² + 8x + 9x + 12 =
6x² + 17x + 2
● Watch the signs● Line up terms, and combine like terms
Multiplication patterns and short cuts:
● (a + b) = a + 2ab + b 2 2 2
Instead of using foil, replace values into formula.
● (a + b) = a + 3a b + 3ab + b^3 3 3 2
the a value decreases left to right, b value increases left to right
5.4 Factor and Solve Polynomial Equations
Rules to follow:
1. Look for common monomials
2. Use factoring Pattern
*difference of squares a^2 b^2 = (a + b) ( a b)
ex. 2x^432x^2
2x^2( x^216)
2x^2 (x + 4) ( x 4)
*sum of cubes a^3 + b^3 = ( a + b) (a^2 ab + b^2)
*diff of cubes a^3 b^3 = ( a b) ( a^2 + ab+ b^2)
ex. 8x^3 125 (something cubed minus something cubed) a= (2x) ^3 b = (5)^3
( 2x 5) ( 4x^2 + 10x + 25)
3. Factorable trinomial ( like what we’ve done with X and super box)
ex. x^2 8x 20
( x 10 ) ( x + 2)
ex. 2x^2 3x + 1
(2x 1) ( x 1)
4. Factor by grouping ( 4 terms break into two different groups, factor out and rewrite, will gettwo sets of () that have same terms)
ex. x^3 7x^2 4x + 28
x^2( x 7) 4( x 7)
(x^2 4 )(x7) = still need to simplify x^2 term because its a difference of squares
(x 2) (x +2)(x7) Set all terms equal to zero to solve
5. Any combination of 1 4
Hints:
• Determine common factors
• Determine common letters
• Pull out the lowest in each term
• Must be in every term, to factor it out
20k – 10k² + 70kz² =
10k( 2 – 5k + 7kz)
● Once the common values are pulled out, determine what is left. 2 terms, use thefactoring patterns. 3 terms use trinomial, 4 terms factor by grouping.
● The degree tells how many solutions there will be
Grouping example:
5.5 Apply the remainder and factor Theorems
Long Division:
● Always look at first term to determine how many times it goes into term in question● If there is a remainder, put it over the divisor as part of the answer● Be sure to place the values you choose, above the appropriate terms● If you are missing a term, account for it anyway by using 0x^value, to keep its place in
answer
Synthetic Division: Only works for linear polynomial divisors
● Division in terms of (x r): Use the opposite value of r.● Write coefficients of each term in order. Use zero if a term is not given.● Add down, multiply divisor diagonally, put new answer under next term.● Continue through last term. The last value is the remainder, put it over the divisor term.● Going from right to left: remainder, constant, coefficents.● Rewrite expression with using coefficients and terms.● Each coefficient will be one degree less than you started with.
2 ways to evaluate a function:
1. Direct Substitution plug values in for x and solve
ex. f(x) = 3x^2 2x + 7 find f(4)
plug 4 in for x and solve.
f(4) = 3(4)^2 2(4) + 7 = 63
2. Synthetic Substitution Follow steps of synthetic division, but do not use opposite of x
f(x) = 3x^2 2x + 7
f(4) 4 3 2 7
12 56
3 14 63 = 63
The remainder is the answer= 63
How to make a general graph of polynomials: f(x) = 2x^5 + 3x^2 + 3
1. Determine if it is odd or even (number of degrees)
2. Determine how many times it will cross the x intercept (same number as degrees)
3. Determine how many turns (one less than the degrees)
4. Determine y intercept (the constant number at the end)
5. Determine what the left and right behavior is (what quadrant it begins in)
● Even positive up / up● Even negative down / down● Odd positive down / up● Odd negative up / down
6. Graph the general line, crossing through the x intercept with certain number of turns andpassing through the y intercept.
Applying synthetic division with story problems/figures:
When given total volume of a shape:
● divide the volume by one of the sides. Factor out to find missing side (use x or super box)
OR● Divide the first answer by the other side. Will get the three sides.● When dividing, you will always get zero for a remainder as they are all factors.
