01: Equivalent Algebraic Expressions and Radicals 

Term: Separated by a + or - sign

Adding and Subtracting Polynomials

1. Expand to remove brackets
2. Collect like terms (terms that are common: x² and x² are like terms but x² and x are not because of the difference in exponents)
   • Arrange final, collected result so that it is in order of highest to lowest exponents (e.g. x³ + x² + x)
   • IF: there is a - sign beside your bracket, you must distribute it in and thus changing the signs of the term(s) in the brackets

Multiplying Polynomials

RULES:

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(a+b)² = a² + 2ab + b²

(a-b)² = a² - 2ab + b²

xⁿ • xᵐ = xⁿ ⁺ ᵐ

(a + b) (a - b) = a² - b²

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Factoring Polynomials

• Always take out a common factor
• 2 terms: look for difference of squares
• 3 terms: look for a quadratic → ax² + bx + c
  • IF: you get a + in your bracket when factoring quadratics, you can’t do anything more

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Simplifying Rational Expressions

Rational expression: a division of polynomials (essentially a fraction) where the denominator can never be = to zero

Stating Restrictions

Restrictions: what cannot happen, what the expression cannot be equal to

1. Take the denominator and make it = to zero (because it is a restriction) → looks like [denominator expression] ≠ 0
2. Isolate to get the variable alone
3. Keep the ≠ sign throughout → this is your new restriction
   • This must be done to every denominator → if you have flipped a rational expression to change your division to multiplication then you must do restrictions on everything that has been a denominator

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Multiplying Rational Expressions

1. Factor the numerator and denominator

2. State restrictions on denominator(s)

3. Reduce first and then multiply, crossing out as needed

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Dividing Rational Expressions

1. Change the multiplication into division by flipping the expression to the right of the = sign

2. Factor

3. State restrictions on both old and new denominators (pre and post flip in step 1)

4. Reduce first and then multiply, crossing out as needed

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Adding and Subtracting Rational Expressions

1. Factor all numerators and denominators
2. State restrictions
3. Reduce if possible
4. Find the Lowest Common Denominator of the fractions
5. Rewrite each term with the Lowest Common Denominator as the denominator, and multiply numerators with same value as you did denominators (what you do to the bottom you must do to the top)
6. Add or subtract the fractions
7. Factor the numerator of the simplified fraction (to check that there can’t be anymore reducing)

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Operations with Radicals

Radical: √

• A radical is any root, the above symbol is used to represent radicals

  Radicand: the term(s) under the radical sign

  Surd: all together (radical and radicand)

Mixed Radicals: the coefficient in front of the surd is a number other than 1 or -1

Radical Rules

• m√a + n√a = (m+n)√a break coefficients up and add, keep one of the √a duplicate surds
• m√a - n√a = (m-n)√a break coefficients up and subtract, keep one of the √a duplicate surds
• √a • √b = √ab merge by multiplying the surds into a single surd
• m√a • n√b = mn√ab merge the coefficients and surds
• √a / √b = √a/b merge the two surds into one fraction with one radical
• m√a / n√b = m/n √a/b split coefficients and surds up
• m(√a + n√b) = m√a + mn√b distribute
• If you multiply a root by its matching exponent, it gets cancelled out (e.g. √2² is 2)

Do not ever put a 2 in the checkmark section of the radical → it is assumed that there is a 2 there and thus it a number is only written if it is not 2 (e.g. a cube root will have a 3 in the checkmark)

Simplifying Radicals

• Must always rationalize the denominator

  • 1 term: take the radical part and multiply it by both numerator and denominator
  • 2 terms: flip the sign and rationalize it out as a difference of squares

  → Make sure that the radicals are the lowest that they can be

• Factor tree, find a coefficient to simplify radicand: look for prime numbers (factor tree) and anything that you have 2 of gets taken out of the radicand and is multiplied by the coefficient → anything that you have only 1 of gets multiplied together stays under the radicand

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