Basic Lessons In Math.docx

Note: All of the Lessons are connected with each other.

Basic Lessons in Math Laws of Exponents:

Special Products:

1.) Product of Powers - For any Real Number a and any positive integers m and n: 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛

1.) Perfect Square Trinomial - Also called the square of a binomial. (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2

2.) Power of a Power - For any real number a and any positive integer m and n: (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛

2.) Difference of Two Squares - It is the product of the sum and difference of two quantities. (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2

3.) Power of a Product - For any real number a and any positive integer m and n: (𝑎𝑏)𝑚 = 𝑎𝑚 𝑏 𝑚

3.) Sum and Differences of Two Cubes: (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 ) = 𝑎3 + 𝑏 3 (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 ) = 𝑎3 − 𝑏 3

4.) Quotient of Powers - For integers m and n, where m > n, and any nonzero number a: 𝑎𝑚 = 𝑎𝑚−𝑛 𝑎𝑛 5.) Power of a Quotient - For real numbers a and b, b≠0, and any positive integer m: 𝑎 𝑚 𝑎𝑚 ( ) = 𝑚 𝑏 𝑏 6.) Negative Exponent - For any nonzero real number a and any positive integer n: 1 𝑎−𝑛 = 𝑛 𝑎 7.) Zero Exponent - Foe any nonzero real number a: 𝑎0 = 1

Multiplying Polynomials: - In multiplying polynomials always use the laws of exponent and the distributive property. Ex. 2𝑥(𝑎 + 𝑏 + 𝑐) 𝐴𝑛𝑠. = 2𝑎𝑥 + 2𝑏𝑥 + 2𝑐𝑥 - In some cases, like two binomials being multiplied, we use a simpler method which is the FOIL Method. Ex. F L (2𝑥 + 𝑦)(4𝑥 + 2𝑦) O I = 8𝑥 2 + 4𝑥𝑦 + 4𝑥𝑦 + 2𝑦 2 𝐴𝑛𝑠. = 8𝑥 2 + 8𝑥𝑦 + 2𝑦 2

Note: All of the Lessons are connected with each other.

4.) Using the Last expression, group the first two terms and the last two terms and factor out of each binomial term. Note that the same binomial factor is considered as only on binomial and the other factored out is to be put together. (4𝑥 2 + 8𝑥) + (−3𝑥 − 6) = 2𝑥(𝑥 + 2) − 3(𝑥 + 2) 𝐴𝑛𝑠. = (2𝑥 − 3)(𝑥 + 2)

Factoring: Factoring Polynomials with a Common Factor: Ex. 4𝑥 2 + 8𝑥 3 + 12𝑥 4 Steps: 1.) Get the common factor of the polynomial.

Factoring; Difference of Two Squares:

2

2

3

4

4𝑥 (4𝑥 + 8𝑥 + 12𝑥 )

Ex.

2.) Divide the common factor to all of the terms. 4𝑥2 +8𝑥3 +12𝑥4 4𝑥 2

4𝑥2

4𝑥 2

= 1 + 2𝑥 + 3𝑥2

3.) Get all of the divided terms and put them in order. 𝐴𝑛𝑠. = 1 + 2𝑥 + 3𝑥 2

Factoring Quadratic Trinomials: Ex.

𝑥2 − 4 Steps: 1.) Get the square root of the terms(not including the signs). √𝑥 2 − √4 =𝑥−2 2.) Get the binomial that has the same terms as step 1 but the second terms sign is the opposite. 𝐴𝑛𝑠. = (𝑥 − 2)(𝑥 + 2)

4𝑥 2 + 5𝑥 − 6 Steps: Sum and differences of Two Cubes

1.) Multiply the coefficient of 𝑥 2 and the constant term. 4(−6) = −24 2.) Find a pair of factors whose product is the number obtained in step 1 and whose sum is the coefficient of 𝑥 or the middle term. 8(−3) = −24 8 + (−3) = 5 3.) Rewrite the equation by replacing the middle term with the pair of numbers with its variable. 4𝑥 2 + 8𝑥 − 3𝑥 − 6

Ex. 𝑥3 − 𝑦3 Steps:

1.) Get the Cube root of the two terms (not including the signs). 3

√𝑥 3 − 3√𝑦 3 = 𝑥 − 𝑦

Note: All of the Lessons are connected with each other.

2.) You already have the first binomial, now lets do the trinomial: a.) Get the square of the first term in the binomial. 𝑥2

2.) Factor them according to how you grouped them. = 2𝑎(𝑥 + 2𝑦) + 𝑏(𝑥 + 2𝑦) 𝐴𝑛𝑠. = (2𝑎 + 𝑏)(𝑥 + 2𝑦)

Rational Expressions: -

b.) Get the opposite product of the two terms in the binomial. 𝑥𝑦 c.) Get the square of the last term in the binomial. 𝑦2

3.) Put all of the terms from a, b, and c to make the trinomial. Put the result of a, b, and c with the result of step 1. 𝐴𝑛𝑠. = (𝑥 − 𝑦)(𝑥 2 + 𝑥𝑦 + 𝑦2)

Factoring by Grouping: Ex. 2𝑎𝑥 + 2𝑏𝑦 + 4𝑎𝑦 + 𝑏𝑥 Steps 1.) Choose Groupings that are easy for you (you can group them into quadratic trinomials, difference of two squares and sum and difference of two cubes depending on the problem). (2𝑎𝑥 + 4𝑎𝑦) + (𝑏𝑥 + 2𝑏𝑦)

-

These are ratio of two polynomials with a denominator not equal to zero. Can be written in the form of 𝑃 𝑄

-

, where P and Q are

polynomials and Q≠0. All parts of a rational expression should be a real number.

Simplifying Rational Expressions: Ex. 3𝑥 2 − 4𝑥 + 1 4𝑥 2 − 8𝑥 + 4 Steps: 1.) Factor the numerator and the denominator. (3𝑥 − 1)(𝑥 − 1) (4𝑥 − 4)(𝑥 − 1) 2.) Cancel out their greatest common factor and copy the remainders. (3𝑥−1)(𝑥−1) (4𝑥−4)(𝑥−1)

𝐴𝑛𝑠. =

3𝑥 − 1 4𝑥 − 4

Note: All of the Lessons are connected with each other.

(𝑥+4)(𝑥−2) 3(𝑥+1)

Multiplying Multiplying Rational Expressions: Ex. 𝑥 + 7 𝑥 2 + 10𝑥 + 25 ∙ 𝑥 2 − 25 𝑥 2 − 49

(𝑥+3)(𝑥+1)

∙

(𝑥−2)

3.) Proceed to multiplying the remaining polynomials. 3𝑥 + 12 𝐴𝑛𝑠. = 𝑥+3

Steps: 1.) Factor out the numerator and denominator of each rational expression. 𝑥+7 (𝑥 + 5)(𝑥 + 5) ∙ (𝑥 − 5)(𝑥 + 5) (𝑥 + 7)(𝑥 − 7) 2.) Cross-Cancel the common factors of the opposite poles. Lastly multiply the remaining polynomials.

(𝑥+5)(𝑥+5) 𝑥+7 ∙ (𝑥−5)(𝑥+5) (𝑥+7)(𝑥−7)

𝑥+5 (𝑥 − 5)(𝑥 − 7) 𝑥+5 𝐴𝑛𝑠. = 2 𝑥 − 12𝑥 + 35 =

Dividing Rational Expression: Ex. 𝑥 2 +2𝑥−8 𝑥 2 +4𝑥+3

÷

𝑥−2 3𝑥+3

Steps: 1.) Get the reciprocal of the divisor.

𝑥 2 + 2𝑥 − 8 3𝑥 + 3 ÷ 𝑥 2 + 4𝑥 + 3 𝑥 − 2 2.) Factor out the numerator and the denominator of both rational expression and cross-cancel the common factor

Good Luck 