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

Study Journal Notes

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Exponents and Radicals:

Definitions:

• Perfect Square

• Square Root

• Perfect Cube

• Cube Root

• Prime Number

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Lesson 1:​​

• ​Determine square roots without a calculator.

  • Know all perfect squares off by heart, at least to 13 x 13 = 169​

• Special Notes:

  • If the index is even and the radicand is negative then the root DOES NOT EXIST!

  • If the index is odd and the radicand is negative then the root is negative.

  • Any number to the 6th power is both a perfect square and a perfect cube.

Lesson 2:​​​​

• Three methods to determine square or cube roots:

  • Draw a diagram: draw a square, find the side length given the area; draw a cube, find the side length given the volume​

  • Prime factorization: know how to make a factor tree

    • know how to group prime factors to determine perfect squares or perfect cubes​

  • Calculator: know your key strokes for radicals (even when the index is bigger than 3).​

• Special Notes:

  • If your radicand is a fraction you can rewrite it as the numerator under the radical over the denominator under the radical (see last example on page 11)​

Lesson 3:​​

• Exponent Rules:​

  • Product of powers​ (add the exponents)

  • Quotient of Powers (subtract the exponents)

  • Power of Powers (multiply the exponents)

  • Zero Exponent (it's always 1)

  • Negative Exponent (base becomes denominator with a positive exponent)

  • Radicals and fractional exponents (the index is the denominator of the rational exponent - that means a fraction)

• Special Notes:

  • Remember the exponent only acts on the base.​

Lesson 4:​​

• Know how to convert a rational (fractional) exponent to a radical (under a root) expression​

  • the denominator of the exponent is the index of the radical​

  • the numerator of the exponent is the new whole exponent of the radicand

• Special Notes:

  • Apply the exponent laws one at a time​

  • Identify the base that the exponent is acting on, make sure you do not accidentally include extra numbers that are not affected by the exponent (those are called coefficients)

Lesson 5:​​

• Know how to add, subtract, and multiply fractions (rational numbers are fractions, and rational exponents are fractional exponents)​

• Rational exponent rules:

  • if any base is raised to the power of (1/n) then take the nth root of the base

  • if any base is raised to the power of (-1/n) then then you take 1 over the nth root of the base

  • if any base is raised to the power of (m/n) then take the nth root of the base to the power of m

  • if any base is raised to the power of (-m/n) then take 1 over the nth root of the base to the power of m

• Special Notes:

  • If you have a question where there is a lot going on (fractional exponents, negatives, rational bases, etc.) choose one thing to work on at a time (i.e. choose to combine like bases first)​

  • Never, ever, ever forget that coefficients (numbers that are not part of a variable base) are still terms raised to the power of 1. That means in the power of a power rule, you need to distribute the power from the outside of the parenthesis to the coefficients as well as the variables.

Lesson 6:​​

• know how to convert from a power to a radical​ and from a radical to a power

• know how to convert from an entire radical to a mixed radical and from a mixed radical to an entire radical

• know how to put irrational numbers in order by converting them to entire radical form

• know how to classify numbers into their appropriate number system (natural, whole, integer, rational, irrational, real)

• Special Notes:

  • For fractional exponents, the denominator of the exponent becomes the index and the numerator becomes the exponent of the radical​

  • Make sure to simplify using your exponent rules as much as possible before converting to radical form...especially when it comes to negative exponents.

Polynomials

Definitions:

• Polynomial

• Real Number

• Whole Number

• Terms

• Exponents

​

Lesson 1:

• Know the general form of a polynomial (real coefficients, whole exponents)

  • Use this knowledge to determine if an expression is a polynomial or not​

• Classify polynomials according to degree (greatest exponent sum of a term of the polynomial), leading coefficient (coefficient of the term with the greatest degree sum), type (by number of terms (terms are separated by + or -) for example: monomial, binomial, trinomial, etc.), and constant (the term with no variables)

• Simplify polynomials by combining like terms (combine the coefficients of terms that have matching variables and variable exponents)

• Special Notes:

  • if a term has more than one variable, add the exponents of all of the variables to determine the degree of the term​

  • put the variables in alphabetical order for all of the terms to determine if you have like terms

Lesson 2:

• Apply the operation (multiplying or dividing) to all items of a term by collecting like items (coefficients, variables with the same base, etc.)

• When distributing a monomial to a polynomial, make sure you distribute to all of the terms of the polynomial. Again, you want to apply the operation (multiplication) to all matching items (coefficients, variables with the same base, etc.)

• Always distribute first, then simplify by combining like terms (see lesson 1 for rules on combining like terms)

• Special Notes:

  • when multiplying or dividing it can be helpful to include a marker "1" for items that are not common to all terms. For example (3xy)(6y + 2x) = (3xy)(6"1"y) + (3xy)(2x"1") = (3)(6)(x)(1)(y)(y) + (3)(2)(x)(x)(y)(1)​ ... remember there should be no variable left behind!

Lesson 3:

• Know how to use algebra tiles to visualize multiplication of polynomials

• FOIL = First (multiply the first term in each binomial), Outside (multiply the outside terms of each binomial), Inside (multiply the inside terms of each binomial) and Last (multiply the last term in each binomial).

• Always simplify by combining like terms after you have completed the FOIL step

• Multiplying two polynomials together gives you the area of a rectangle with polynomial side lengths

Lesson 4:

• Recognize that FOIL can be used to solve area problems if the dimensions of a rectangle are given as binomials

• The distributive property can be used for binomials multiplied by trinomials --> multiply each term of the binomial to each term of the trinomial, then combine like terms

Lesson 5:

• Know the definition of a greatest common factor for a polynomial (see your notes section 5.2)

• remember for polynomials that a greatest common factor must apply to all terms in the polynomial

• if you have a polynomial and all of the terms have a common factor, you can divide the common factor out of each term, write the common factor outside of a set of parentheses and keep the "leftovers" or quotient inside of the parentheses.

• When you have a grouping of common factors and quotients ("leftovers" in parentheses) and the quotients are the same, you can group the quotients together and group the factors together

  • for example: 7x(x - 3) + 4(x - 3)​ = (x - 3)(7x + 4)

• If you have a polynomial with four terms then you can factor the polynomial by grouping the terms and finding a common factor for each pair of terms

  • for example: 2ac + 6ab + 5bc + 15^2 = 2a(c + 3b) + 5b(c + 3b)​ = (2a + 5b)(c + 3b)

Lesson 6:

• Factoring trinomials (THIS IS THE MOST IMPORTANT THING) - follow the steps:

  • Factor out the GCF (if there is one)​

  • find two numbers that will have a product (multiply) of "ac" (the product of the leading coefficient and the constant) and have a sum (add) of b (the coefficient of the middle term)

  • rewrite the trinomial as four terms (break the middle term up into the two numbers you found in step two)

  • Factor by grouping (see lesson 5)

  • rewrite your trinomial in factored form **don't forget the GCF if there is one**

Lesson 7:

• remember when factoring trinomials to look for a GCF first

• remember when factoring trinomials that the two numbers you are looking for must multiply to give you the product of ac (the product of the leading coefficient and the constant)

Lesson 8:

• Be able to identify a difference of squares (a perfect square minus a perfect square). BE CAREFUL sometimes you will have to factor out a GCF in order to see the difference of squares. The factors of a difference of squares are the difference of the roots of the squares multiplied by the sum of the roots of the squares (see notes page 25)

• Be able to identify a perfect square trinomial (made by squaring a single binomial). The coefficients "a" and "c" will be perfect squares and the coefficient "b" will be double the product of the roots of "a" and "c" (see notes page 26)