Algebraic expressions

• Algebraic expressions are a single number or a combination of numbers and variables multiplied together.
• Algebraic expressions do not have an equal sign and terms are separated by a addition or subtraction sign.
• Monomial is a polynomial with only one term, e.g. 5x
• Binomials are polynomials that contain two terms, e.g. x + y
• Trinomials are polynomials that contain three terms, e.g. x - 2y + z
• Polynomials can be added or subtracted using the distributive law.
• Polynomials can be added to other polynomials using the distributive law.
• The distributive law states that when multiplying a bracket, we distribute the factor outside the brackets to all the terms inside the brackets.
• Product of two binomials: a method known as FOIL can be used to multiply two binomials
• FOIL:

  First

  Outer

  Inner

  Last

  and in order.
• Binomials squared:

  e.g: (7m+4n)^2

  1. square the first term
  2. multiple what is inside the bracket
  3. square the last term
• Two binomials that are the same but have different signs:

  e.g: (7m+4n)(7m-4n)

  1. square the first term
  2. square the last term
  3. separate it with a negative sign
• When finding the product of three or more binomials, we use the distributive property repeatedly until there are only single variables left.
• If x is raised to an even power, the result will always be positive regardless of whether x is positive or negative.

  
• Sum and difference of two cubes:

  e.g: x^3+y^3

  1. Cube the both terms, then close the bracket
  2. Square the first term (from the bracket in step 1)
  3. The sign should be the opposite of the sign in the first bracket
  4. Multiply both terms
  5. Always add a positive sign
  6. Square the last term

  Note: exponent must be divisible by three to be a cube.

  
• (a^3 + b^3) = (a+b)(a^2 - ab + b^2)
• (a^3 - b^3) = (a-b)(a^2 + ab + b^2)
• Difference of two squares (DOTS):

  e.g: 25x^2-y^4

  
  

  1. Square the coefficient
  2. Square the variables (exponents)
  3. Then repeat if necessary

  Note: this can only be done when there is a subtraction sign.
• Taking out a high common factor with DOTS:

  e.g:2x^2-8

  
  

  1. Look for the HCF of the coefficient and constant
  2. Place it outside the bracket
  3. Place the remains of the sum inside
  4. DOTS
• Factorization is the reverse operation of the distributive law
• Highest Common Factor:

  e.g: 3a^2+ab+2a^3

  1. Look for the HCF of the coefficients then the variables
  2. Take out the variables with the lowest exponent
• Sign Change:

  e.g: x(a-b)+2(b-a)

  
  

  1. Put a positive/negative sign outside the term
  2. Swap the numbers inside the bracket around
  3. Keep the negative sign in front
  4. Factorise

  Note: this is only done with terms that have a negative sign

  (it does not matter if there is a positive sign)

  
  

  Note: the negative/positive sign applies to both brackets but it can only affect one bracket at a time.
• Advanced Trinomials consist of two methods

  1. XL Method - regular trinomials
  2. X Method - advanced trinomials
• The XL Method:

  e.g: x^2+7x+12

  1. Draw an X then write the factors of the first term on the left side.
  2. Write down the factors of the last term that will give you the middle term when added/subtracted on the right side.
  3. Multiply all the factors following the X shape.
  4. Draw an L then add/subtract the factors for the middle term
  5. Determine the signs needed to get the same sign as the middle term.
  6. Factorise
• The X Method:

  e.g: 21x^2+25-4

  1. Draw an X, multiply the first and the last terms put it at the top of the X.
  2. Write the factors of the product of step 1 that will give you the middle term when added/subtracted on the sides.
  3. Place the middle term at the bottom.
  4. Determine the signs needed to get the middle term sign.
  5. Return to sum, split the middle term into the founded numbers and place the variable on both numbers.
  6. Group the two sums, look for HCF.
  7. Factorise
• Grouping:

  e.g: ab+3a+2b+6

  1. Take out the HCF of the two grouped terms.
  2. Place a bracket around the remaining sum.
  3. Take the common bracket and place the remaining numbers in a bracket
• Simplification of Algebraic Fractions

  
  

  1. Multiplication and Division
  2. Addition and Subtraction
• Multiplication and Division of Fractions

  
  

  1. Factorise the each numerator and denominator.
  2. Cross cancel
  3. Simplify

  Note: The division and multiplication sign make it one fraction

  
  

  Note: Tip and Time & All or Nothing
• Addition and Subtraction of fractions

  1. Factorise each numerator and denominator.
  2. Multiply by a Lowest Common Denominator (LCD)
  3. Cross cancel
  4. Simplify

  Note: when we have the LCD, take out the highest exponent

  
  

  Note: the addition and subtraction sign at the numerator will not allow any of the terms to be divided by the denominator.

  
  

  You cannot remove/change a bracket so we add one.
• The Real Number System consists of rational numbers and irrational numbers
• Rational numbers are numbers that can be expressed as a ratio of integers and are subdivided into two main types: fractions and integers

  • Positive Integers: Z (e.g: 0, 1, 4)
  • Negative Integers: Z- (e.g: -1, -5,-68)
  • Whole Numbers: N0 (e.g: 0, 3, 45)
  • Natural Numbers: N (e.g: 1, 6, 32)
  • Fractions: F (rational numbers that are not integers)
• Decimal Fractions are used to represent fractions with a decimal point.

  Terminating Decimals:

  a decimal that ends (finite number of digits)

  Recurring Decimals:

  a decimal that has an infinite pattern of the same number(s)
• Irrational Numbers are numbers that cannot be expressed as a ratio of integers and are subdivided into two main types: non-terminating and non-recurring decimals.

  
  

  Non-terminating Decimals:

  a decimal that never ends (infinite numbers of digits)

  Non-recurring Decimals:

  a decimal that has infinite amount different numbers
• A term is an expression with one or more variables, raised to a power (exponent) and/or multiplied by a constant.