Simplifying and Factoring Polynomials

• Polynomials are algebraic expressions containing variables and coefficients
• Variables have non-negative integer exponents
• Simplifying polynomials involves combining like terms, which have the same variable raised to the same power
• Factoring polynomials is the reverse process of expanding them
• Factoring breaks down a polynomial into a product of simpler polynomials or constants

Simplifying Polynomials

• Combine like terms by adding or subtracting their coefficients
• Like terms have the same variable raised to the same power
• 3x^2 + 5x^2 simplifies to 8x^2
• Distribute multiplication over addition or subtraction
• 2x(x + 3) expands to 2x^2 + 6x
• Use the order of operations (PEMDAS/BODMAS) to simplify expressions
• Remove parentheses by distributing any coefficients or signs
• Add or subtract polynomials by combining like terms
• Multiply polynomials by distributing each term of one polynomial to each term of the other polynomial
• Use FOIL (First, Outer, Inner, Last) method for multiplying two binomials
• (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6

Factoring Polynomials

• Factoring expresses a polynomial as a product of two or more polynomials or monomials
• Identify the greatest common factor (GCF) of all terms in the polynomial
• Factor out the GCF from each term
• To factor 6x^2 + 9x, the GCF is 3x, so 6x^2 + 9x = 3x(2x + 3)
• Recognize and factor difference of squares: a^2 - b^2 = (a + b)(a - b)
• To factor x^2 - 4, this is a difference of squares: (x + 2)(x - 2)
• Recognize and factor perfect square trinomials: a^2 + 2ab + b^2 = (a + b)^2 and a^2 - 2ab + b^2 = (a - b)^2
• To factor x^2 + 6x + 9, this is a perfect square trinomial: (x + 3)^2
• Factor quadratic trinomials of the form ax^2 + bx + c
• Find two numbers that multiply to 'ac' and add up to 'b'
• Rewrite the middle term using these two numbers
• Factor by grouping
• To factor x^2 + 5x + 6, find two numbers that multiply to 6 and add to 5 (2 and 3)
• Rewrite as x^2 + 2x + 3x + 6
• Factor by grouping: x(x + 2) + 3(x + 2) = (x + 2)(x + 3)
• For polynomials with more than three terms, try factoring by grouping
• Group terms in pairs and factor out the GCF from each pair
• If the remaining binomial factors are the same, factor it out from the entire expression

Applications of Factoring

• Solving polynomial equations involves setting the polynomial equal to zero and factoring
• Set each factor equal to zero and solve for the variable
• To solve x^2 - 4 = 0, factor to (x + 2)(x - 2) = 0
• Solutions are x = -2 and x = 2
• Simplifying rational expressions means factoring the numerator and denominator and cancel common factors
• Combining rational expressions requires finding a common denominator by factoring the denominators
• Identifying discontinuities in rational functions involves factoring the denominator to find values that make it zero

Polynomial Degree

• The degree of a term in a polynomial is the sum of the exponents of the variables in that term
• The degree of a polynomial is the highest degree of any term in the polynomial
• A constant term has degree 0 (e.g., 5 has degree 0)
• A linear term has degree 1 (e.g., 3x has degree 1)
• A quadratic term has degree 2 (e.g., 2x^2 has degree 2)
• A cubic term has degree 3 (e.g., 4x^3 has degree 3)
• The degree of a polynomial affects its end behavior and the maximum number of turning points
• A polynomial of degree n has at most n-1 turning points
• The leading coefficient is the coefficient of the term with the highest degree
• The leading coefficient and degree determine the end behavior of the polynomial
• Even-degree polynomials have the same end behavior on both sides (both up or both down)
• Odd-degree polynomials have opposite end behavior on each side (one up, one down)
• A positive leading coefficient for an even-degree polynomial means both ends go up
• A negative leading coefficient for an even-degree polynomial means both ends go down
• A positive leading coefficient for an odd-degree polynomial means it goes down on the left and up on the right
• A negative leading coefficient for an odd-degree polynomial means it goes up on the left and down on the right