3. Power of a Product: Separate letters, distribute to the exponents and the bases


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1 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
2 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 * 10 ³ = Positive powers, start big Ex * 10³ = 1250 Exponential Growth: A = C( 1 + r)t Exponential Decay: A = C(1 r)t
3 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 term singlely. With multiple terms, just distribute and combine like terms.
4 (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: 2 (a + b) = a 2 + 2ab + b 2 Instead of using foil, replace values into formula. 3 (a + b) = a 3 + 3a 2 b + 3ab + b^3 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^4 32x^2 2x^2( x^2 16) 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)
5 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 get two sets of () that have same terms) ex. x^3 7x^2 4x + 28 x^2( x 7) 4( x 7) (x^2 4 )(x 7) = still need to simplify x^2 term because its a difference of squares (x 2) (x +2)(x 7) 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 the factoring patterns. 3 terms use trinomial, 4 terms factor by grouping. The degree tells how many solutions there will be Grouping example:
6 5.5 Apply the remainder and factor Theorems Long Division:
7 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
8 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.
9 f( 4) = 3( 4)^2 2( 4) + 7 = Synthetic Substitution Follow steps of synthetic division, but do not use opposite of x f(x) = 3x^2 2x + 7 f( 4) = 63 The remainder is the answer= 63 How to make a general graph of polynomials: f(x) = 2x^5 + 3x^ 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 and passing 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)
10 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.
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