Algebra, Indices & Probability Study Guide

Algebra

Binomial Expansion

• Expanding $(a + b)^2$:
  • Formula: $(a + b)^2 = a^2 + 2ab + b^2$
    • This formula shows how to expand the square of a binomial, resulting in a trinomial.
  • Example: $(x + 5)^2 = x^2 + 10x + 25$
    • Here, $a = x$ and $b = 5$, so the expansion follows the formula.
• Expanding $(a - b)^2$:
  • Formula: $(a - b)^2 = a^2 - 2ab + b^2$
    • This is similar to the previous expansion but with a subtraction.
• Expanding $(a + b)(a - b)$:
  • Formula: $(a + b)(a - b) = a^2 - b^2$ (Difference of Two Squares)
    • This expansion results in the difference of two squares, a useful pattern in algebra.

Difference of Two Squares

• Recognize the form: $a^2 - b^2$
  • This pattern is characterized by two perfect squares separated by a subtraction sign.
• Factorize as: $(a + b)(a - b)$
  • This is the reverse of the expansion, allowing factorization of the difference of two squares.
• Example: $x^2 - 25 = (x + 5)(x - 5)$
  • Here, $a = x$ and $b = 5$, so the factorization follows the formula.

Perfect Squares

• Recognize trinomials like: $x^2 + 6x + 9$
  • Perfect square trinomials can be factored into the square of a binomial.
• Factorize as: $(x + 3)^2$ (since $6 = 2 Imes 3$ and $9 = 3^2$)
  • This example demonstrates factoring a perfect square trinomial.
• Patterns to remember:
  • $a^2 + 2ab + b^2 = (a + b)^2$
  • $a^2 - 2ab + b^2 = (a - b)^2$
    • These are the two primary patterns for perfect square trinomials.

Algebraic Techniques

Collecting Like Terms

• Combine terms with the same variable and exponent.
  • This involves adding or subtracting coefficients of like terms.
• Example: $2x + 5x = 7x$
  • Here, both terms have the same variable $x$, so their coefficients can be added.

Expanding Brackets

• Use distributive property:
  • $a(b + c) = ab + ac$
    • This property allows you to multiply a term by each term inside the brackets.
  • Example: $2(x + 3) = 2x + 6$
    • Here, 2 is distributed to both $x$ and 3.

Factorising

• Take out common factors or use identities:
  • Example: $3x + 6 = 3(x + 2)$
    • Here, 3 is a common factor in both terms.
  • Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$
    • This is an example of factorizing a quadratic expression.

Solving Linear Equations

• One Step: $x + 5 = 9 othe x = 4$
  • These equations require one step to isolate the variable.
• Two Step: $2x + 3 = 7 othe 2x = 4 othe x = 2$
  • These equations require two steps to isolate the variable.
• With Brackets: $3(x - 2) = 9 othe 3x - 6 = 9 othe 3x = 15 othe x = 5$
  • These equations involve expanding brackets before solving.

Equations with Fractions & Cross Multiplying

• If: $a/b = c/d$, then $a Imes d = b Imes c$
  • Cross multiplication is a technique to solve equations involving fractions.
• Example: $4/x = 2/3 othe 2x = 12 othe x = 6$
  • Here, cross multiplication helps to isolate the variable. 2. Indices (Exponents)

Indices (Exponents)

Laws of Indices

• Product Rule: $a^m Imes a^n = a^{m+n}$
  • When multiplying like bases, add the exponents.
• Quotient Rule: $a^m / a^n = a^{m-n}$
  • When dividing like bases, subtract the exponents.
• Power of a Power: $(a^m)^n = a^{mn}$
  • When raising a power to a power, multiply the exponents.
• Zero Index: $a^0 = 1$
  • Any non-zero number raised to the power of 0 is 1.
• Negative Index: $a^{-n} = 1 / a^n$
  • A negative exponent indicates a reciprocal.
• Fractional Index: a^{1/n} = \sqrt[n]{a}
  • A fractional exponent indicates a root.

Simplifying Expressions

• Combine like bases, apply laws
  • Simplify exponential expressions by using the laws of indices.
• Example: $2^3 Imes 2^5 = 2^8$
  • Applying the product rule.

Probability

Basic Probability & Complementary Events

• Probability of event A: $P(A) = \frac{\text{Number of favourable outcomes}}{\text{Total outcomes}}$
  • This is the basic formula for calculating probability.
• Complement: $P(A') = 1 - P(A)$
  • The probability of an event not occurring is 1 minus the probability of it occurring.
• Example: $P(\text{rolling a 4}) = 1/6$
  • A standard six-sided die has one face with a 4.

Venn Diagrams

• Sets: Represented by circles
  • Venn diagrams use circles to represent sets and their relationships.
• Union ($A Imes B$): A or B or both
  • The union includes all elements in either set A or set B or both.
• Intersection ($A Imes B$): A and B
  • The intersection includes only elements in both sets A and B.
• Complement ($A'$): Not A
  • The complement includes all elements not in set A.
• Tip: Fill in known values first
  • When solving problems with Venn diagrams, start with the known values.

Two-Way Tables

• Used to organize data (e.g. gender vs preference)
  • Two-way tables can organize data to show relationships between two categorical variables.
• Add row and column totals
  • Calculate marginal probabilities
• Use table to find probabilities
  • The table can be used to find probabilities of different events.
• Example: Probability of choosing a girl who likes football?
  • Use the counts in the table to compute conditional probability.

Practice Tips

• Show all working for full marks
  • Always show each step in your solution to ensure you receive full credit.
• Check answers by substituting back
  • Verify your solution by plugging it back into the original equation.
• Draw diagrams clearly (Venn, tables)
  • Make sure your diagrams are neat and labeled clearly.
• Practice with past exam questions
  • Use past papers to prepare for the exam.