Expanding Brackets and Distributive Property

• Expanding brackets involves multiplying each term inside the bracket by the term outside
• Simplifies expressions and helps in solving equations

Distributive Property

• A fundamental concept in algebra
• a(b + c) = ab + ac, where a, b, and c are numbers or variables
• Expanding brackets is an application of the distributive property
• 3(x + 2) = 3x + 6
• -2(y - 5) = -2y + 10
• The distributive property can be extended to more complex expressions
• a(b + c + d) = ab + ac + ad
• When expanding brackets, pay attention to signs
• Multiplying by a negative number changes the sign of each term inside the bracket
• -a(b - c) = -ab + ac

Binomial Expansion

• Involves expanding expressions of the form (a + b)^n
• For small values of n, direct multiplication can be used.
• (a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2
• (a + b)^3 = (a + b)(a + b)(a + b) = a^3 + 3a^2b + 3ab^2 + b^3
• Pascal's triangle provides coefficients for binomial expansion
• The binomial theorem gives a general formula for expanding (a + b)^n
• (a + b)^n = Σ [n! / (k!(n-k)!)] * a^(n-k) * b^k, where k ranges from 0 to n
• (x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
• The coefficients in the expansion are binomial coefficients
• The binomial coefficients can be calculated using combinations
• The coefficient of the term a^(n-k)b^k is often written as "n choose k" or C(n, k)

Factoring Expressions

• Is the reverse process of expanding brackets
• Involves finding the factors that multiply together to give the original expression
• Simplest type of factoring is finding the greatest common factor (GCF)
• 6x + 9 = 3(2x + 3), where 3 is the GCF
• Factoring quadratic expressions involves finding two binomials that multiply to give the quadratic
• x^2 + 5x + 6 = (x + 2)(x + 3)
• Difference of squares: a^2 - b^2 = (a + b)(a - b)
• Perfect square trinomial: a^2 + 2ab + b^2 = (a + b)^2
• Factoring by grouping is used for expressions with four or more terms
• ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
• Factoring is useful for simplifying expressions and solving equations

Simplifying Polynomials

• Involves combining like terms
• Like terms have the same variable raised to the same power
• 3x^2 + 2x - x^2 + 5x = 2x^2 + 7x
• Polynomials can be simplified by adding, subtracting, multiplying, or dividing them
• Adding polynomials involves combining like terms
• Subtracting polynomials involves distributing the negative sign and then combining like terms
• Multiplying polynomials involves using the distributive property
• Dividing polynomials can be done using long division or synthetic division
• Simplifying polynomials makes it easier to work with them
• Simplified polynomials are easier to evaluate and graph

Solving Equations

• Involves finding the value(s) of the variable that make the equation true
• The goal is to isolate the variable on one side of the equation
• Expanding brackets can be a necessary step in solving equations
• 2(x + 3) = 10 becomes 2x + 6 = 10, then 2x = 4, and finally x = 2
• Factoring can also be used to solve equations, especially quadratic equations
• x^2 + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0, so x = -2 or x = -3
• The zero product property states that if ab = 0, then a = 0 or b = 0
• This property is used when solving equations by factoring
• Equations can have one solution, no solutions, or infinitely many solutions
• Linear equations have one solution
• Quadratic equations can have two, one, or no real solutions
• Equations with no solutions are called contradictions
• Equations with infinitely many solutions are called identities
• When solving equations, it is important to check the solutions
• Checking the solutions ensures that they satisfy the original equation
• Extraneous solutions can arise when solving certain types of equations
• Extraneous solutions are solutions that do not satisfy the original equation