Algebraic expressions and factoring are key building blocks in algebra. They help us simplify complex math problems and find solutions more easily. Understanding these concepts is crucial for tackling more advanced math topics.

Polynomials, their operations, and factoring techniques are essential skills in algebra. These tools allow us to solve equations, graph functions, and analyze real-world scenarios. Mastering these concepts opens doors to higher-level math and problem-solving abilities.

Polynomial classification

Polynomial definition and components

• A polynomial is an expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables
• The degree of a polynomial is the highest degree of its terms, where the degree of a term is the sum of the exponents of the variables that appear in it
• The leading coefficient of a polynomial is the coefficient of the term with the highest degree
• The standard form of a polynomial is to write the terms by descending degree, left to right

Classification by degree and number of terms

• Polynomials are classified by degree (linear, quadratic, cubic) and by number of terms (monomial, binomial, trinomial)
• Linear polynomials have a degree of 1 ($ax + b$)
• Quadratic polynomials have a degree of 2 ($ax^2 + bx + c$)
• Cubic polynomials have a degree of 3 ($ax^3 + bx^2 + cx + d$)
• Monomials consist of a single term ($3x^2$, $-5y$)
• Binomials consist of two terms ($x + 2$, $3x^2 - 4y$)
• Trinomials consist of three terms ($2x^2 + 3x - 1$, $x^3 - 2x + 5$)

Polynomial operations

Addition and subtraction

• Adding polynomials involves combining like terms, which are terms with the same variables and exponents
• Subtracting polynomials involves distributing the negative sign to each term of the polynomial being subtracted, then combining like terms
• Example: $(3x^2 + 2x - 1) + (2x^2 - 3x + 4) = 5x^2 - x + 3$
• Example: $(4x^3 - 2x + 1) - (x^3 + 3x - 2) = 3x^3 - 5x + 3$

Multiplication and division

• Multiplying polynomials involves applying the distributive property and the FOIL method (First, Outer, Inner, Last) for multiplying binomials
• The product of two polynomials will have a degree equal to the sum of the degrees of the factors
• Polynomial long division is a method for dividing a polynomial by another polynomial of equal or lower degree
• Example: $(2x + 3)(x - 1) = 2x^2 + x - 3$
• Example: $(x^2 + 3x - 4) \div (x + 4) = x - 1$

Polynomial factoring

Greatest common factor and grouping

• Factoring a polynomial is the process of expressing it as a product of lower-degree polynomials
• The greatest common factor (GCF) of a polynomial is the largest factor that divides all terms of the polynomial
• Factoring by grouping involves arranging the terms into groups that have a common factor, factoring out the GCF from each group, and then factoring the resulting expression
• Example: $12x^3 + 18x^2 = 6x^2(2x + 3)$
• Example: $2x^2 + 5x + 3x + 6 = (2x^2 + 3x) + (5x + 6) = x(2x + 3) + 3(2x + 3) = (x + 3)(2x + 3)$

Special factoring patterns

• Special factoring patterns include the difference of squares ($a^2 - b^2 = (a+b)(a-b)$), perfect square trinomials ($a^2 \pm 2ab + b^2 = (a \pm b)^2$), and the sum or difference of cubes ($a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$)
• Factoring a quadratic expression ($ax^2 + bx + c$) depends on the values of $a$, $b$, and $c$. If $a = 1$, the expression can be factored as $(x + m)(x + n)$ where $m + n = b$ and $mn = c$
• Example: $x^2 - 9 = (x + 3)(x - 3)$
• Example: $x^2 + 6x + 9 = (x + 3)^2$
• Example: $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$
• Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$

Solving quadratic equations by factoring

Quadratic equations and the zero-product property

• A quadratic equation is an equation that can be written in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$
• The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero
• To solve a quadratic equation by factoring: (1) Write the equation in standard form, (2) Factor the left side of the equation, (3) Apply the zero-product property to solve for the variable, (4) Check the solutions by substituting them back into the original equation

Solutions and graphical representation

• The solutions (roots) of a quadratic equation are the $x$-intercepts of the corresponding quadratic function
• A quadratic equation can have 0, 1, or 2 real solutions, depending on the values of $a$, $b$, and $c$, and the discriminant ($b^2 - 4ac$)
• If the discriminant is positive, the quadratic equation has two distinct real solutions
• If the discriminant is zero, the quadratic equation has one repeated real solution
• If the discriminant is negative, the quadratic equation has no real solutions (two complex solutions)
• Example: $x^2 - 5x + 6 = 0 \rightarrow (x - 2)(x - 3) = 0 \rightarrow x = 2$ or $x = 3$
• Example: $x^2 - 2x + 1 = 0 \rightarrow (x - 1)^2 = 0 \rightarrow x = 1$