"The only way to be totally free is through education" - Jose Marti

Study Journal Notes

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Polynomials

Lesson 1:

• Division of polynomials by long division

  • Similar to long division for integers: go one term at a time; line up the term of the quotient with the similar term of the dividend

  • Remember to fill in any missing degree terms with zero

  • Remember to distribute your negative sign when subtracting

• Division of polynomials by synthetic division

  • Use only coefficients and your “a” value

  • zig-zag : add down, multiply across to the new column

  • Your result starts with the coefficient one degree lower than the highest degree of your polynomial

  • Remember to fill in any missing degree terms with zero

Lesson 2:

• Remainder theorem - if a polynomial, P(x), is divided by (x-a) then the remainder is equal to P(a)

• Know how to find a missing coefficient if you are given a polynomial, a divisor, and a remainder

Lesson 3:

• Factor Theorem - if P(a) = 0 then x-a is a factor of P(x)

• How to find possible factors if you’re only given the polynomial

  • Check the factors of the constant as potential “a” values, since a factor of (x-a) means that “a” must be a factor of the constant (but not vice-versa!)

Lesson 4:

• What is a polynomial? - need to have whole number exponents, and the coefficients have to be real numbers

  • Remember y = n (where n = a real number) is still a polynomial!​

• Properties of graphs with odd degrees:

  • positive leading coefficient - left side goes down to negative infinity (Q3), right side goes up to positive  infinity​ (Q1)

  • negative leading coefficient - left side foes up to positive infinity (Q2), right side goes down to negative infinity (Q4)

• Properties of graphs with even degrees:

  • positive leading coefficient - both ends go up to positive infinity (Q2 to Q1 trend)​

  • negative leading coefficient - both ends go down to negative infinity (Q3 to Q4 trend)

• To draw a graph, find all relevant points (x-intercept(s), y-intercept, local maximum(s), local minimum(s)) and draw following the graph properties above from left to right.

• The domain and range of polynomials consists of all real numbers, but the range of even-degree polynomials is limited by the global maximum or global minimum of the graph.

Lesson 5:

• Know the relationship between zeros, x-intercepts, and roots (the "a" value of a factor is an x-intercept, which is a root of the polynomial)

• Know how to find x-intercepts from a factored polynomial --> use this to graph the function

• Given the graph of a function, know how to find the factors from the observed intercepts and their behavior (multiplicity)

  • once you have the factored form, represent the leading coefficient as a variable and substitute in a known point from the graph to find the value of the leading coefficient.​

• Multiplicity --> the number of times that a root shows up tells us about the behavior of the function at the related x-intercept

  • multiplicity of 1 - the function crosses the x-axis at this point​

  • multiplicity of 2 - the function touches the x-axis at this point but does not cross it - true for any even-degree factor

  • multiplicity of 3 - the function crosses the x-axis at this point and acts as an inflection (s-curve) - true for any odd degree factor greater than or equal to 3

Lesson 6:

• When you are solving a problem, try to draw a diagram/picture first

• Read the question carefully to determine what method you should use to solve the problem (graphically vs. algebraically) - the method that you do not use can be applied as a "check"

• When you need to find the leading coefficient of a polynomial that you have factored, choose a coordinate pair that you do know and substitute the values in for "x" and "y" to solve for "a"

• Be careful to consider impossible values (think of the box in the second example, the dimensions cannot possibly be negative, so that narrowed down our choices)

Links to help you study:

• Synthetic Division:

  • http://www.mesacc.edu/~scotz47781/mat120/notes/divide_poly/synthetic/synthetic_division_practice.html​

• Long Division​​:

  • https://study.com/academy/practice/quiz-worksheet-polynomial-long-division.html

  • http://www.mesacc.edu/~scotz47781/mat120/notes/divide_poly/long_division/long_division_practice.html

• Remainder Theorem and Factor Theorem:

  • https://study.com/academy/practice/quiz-worksheet-applying-the-remainder-theorem-factor-theorem.html​

  • https://worksheets.tutorvista.com/remainder-theorem-worksheet.html

• Polynomial graphing characteristics:

  • http://www.softschools.com/quizzes/algebra/identifying_polynomial_or_not_polynomial/quiz4373.html​

  • https://www.vitutor.com/alg/polynomials/polynomials_worksheets.html

  • http://www.shelovesmath.com/algebra/advanced-algebra/graphing-polynomials/

  • http://www.analyzemath.com/math_problems/polynomial_problems.html

• Polynomial Word Problems:

  • https://braingenie.ck12.org/skills/106896

  • https://braingenie.ck12.org/skills/106914

 

Radicals and Rationals

Lesson 7:

• Given a function f(x) and a function that is the square root of f(x), make a table of values, graph the functions and define the domain and range of each function

• Know that there are limitations to the behavior of the square root of a function depending on the value of the function at a particular point (for example, where f(x) < 0, the root of f(x) is undefined)

• Know how to solve for the invariant points given a graph of f(x)

  • set f(x) equal to 1, solve for x​

  • set f(x) equal to 0, solve for x

  • inspect the graph for x-intercepts (where f(x) =0)​​

Lesson 8:

• Identify restrictions (f(x) must be greater than or equal to 0)

• Graph radicals in two ways:

  • let y1 = LHS, let y2 = RHS and look for the intersection between the two points​

  • move all terms to the LHS, make RHS = 0, let y1 = LHS and look for the x-intercepts

• Solve radicals algebraically by rearranging the equation and isolating for x (don't forget to square in order to get rid of the square root)

• verify your claims --> always use substitution in order to check your answer!

Lesson 9/10:

• When dealing with a rational polynomial your first step is to factor...Always factor first!

• Non-permissible values could be vertical asymptotes or points of discontinuity​

  • look at the denominator, what is x not allowed to be? (make the denominator equal to zero and solve for x, since we are never allowed to divide by zero)​

  • Can you simplify any factors to one? (top and bottom cancel) --> you have a point of discontinuity at the zero of the factor

  • Unable to simplify factors to one (can't cancel anything)? --> all of your NPV's are vertical asymptotes

• Inspect the degree of the numerator and the denominator to find horizontal asymptotes

  • If the degree of the denominator is greater than the degree of the numerator --> you have a H.A. at 0​

  • If the degree of the denominator is the same as the degree of the numerator --> you have a H.A. at the fraction made by the leading coefficients of the numerator and the denominator

  • If the degree of the denominator is less than the degree of the numerator --> you do not have a H.A.

Lesson 11:

• Solve rational polynomials graphically

  • let y1 = LHS, let y2 = RHS and look for the intersection between the two points​

  • move all terms to the LHS, make RHS = 0, let y1 = LHS and look for the x-intercepts

• Solve rational polynomials algebraically by rearranging and isolating for x

• verify your claims --> always use substitution to check your answer!