01: Equivalent Algebraic Expressions and Radicals

Term: Separated by a **+** or **-** sign

# Adding and Subtracting Polynomials

Expand to remove brackets

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:

(a+b)² = a² + 2ab + b²

(a-b)² = a² - 2ab + b²

xⁿ • xᵐ = xⁿ ⁺ ᵐ

(a + b) (a - b) = a² - b²

# 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

# 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

Take the denominator and make it = to zero (because it is a restriction) → looks like [denominator expression] ≠ 0

Isolate to get the variable alone

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

# Multiplying Rational Expressions

Factor the numerator and denominator

State restrictions on denominator(s)

Reduce first and then multiply, crossing out as needed

# Dividing Rational Expressions

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

Factor

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

Reduce first and then multiply, crossing out as needed

# Adding and Subtracting Rational Expressions

Factor all numerators and denominators

State restrictions

Reduce if possible

Find the Lowest Common Denominator of the fractions

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)

Add or subtract the fractions

Factor the numerator of the simplified fraction (to check that there can’t be anymore reducing)

# 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

Term: Separated by a **+** or **-** sign

# Adding and Subtracting Polynomials

Expand to remove brackets

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:

(a+b)² = a² + 2ab + b²

(a-b)² = a² - 2ab + b²

xⁿ • xᵐ = xⁿ ⁺ ᵐ

(a + b) (a - b) = a² - b²

# 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

# 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

Take the denominator and make it = to zero (because it is a restriction) → looks like [denominator expression] ≠ 0

Isolate to get the variable alone

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

# Multiplying Rational Expressions

Factor the numerator and denominator

State restrictions on denominator(s)

Reduce first and then multiply, crossing out as needed

# Dividing Rational Expressions

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

Factor

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

Reduce first and then multiply, crossing out as needed

# Adding and Subtracting Rational Expressions

Factor all numerators and denominators

State restrictions

Reduce if possible

Find the Lowest Common Denominator of the fractions

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)

Add or subtract the fractions

Factor the numerator of the simplified fraction (to check that there can’t be anymore reducing)

# 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