Factoring polynomials is a key skill in algebra, breaking down complex expressions into simpler parts. It's like solving a puzzle, finding the building blocks that make up a larger equation. This process helps simplify and solve equations, making it easier to understand their behavior.
From finding the to tackling special forms like perfect squares, there are various techniques to master. These methods not only simplify polynomials but also reveal important information about their roots and structure, laying the groundwork for more advanced algebraic concepts.
Factoring Polynomials
Greatest common factor of polynomials
Largest factor that divides all terms in a polynomial without leaving a remainder
Includes both coefficients and variables (6x2y and 3xy2 have GCF of 3xy)
Extract GCF by factoring it out of each term ax+bx=x(a+b)
6x2y+3xy2=3xy(2x+y) factors out GCF of 3xy
Methods for factoring trinomials
Factor trinomials in the form ax2+bx+c by finding factors of ac that add up to b
2x2+7x+3=(2x+1)(x+3) since 2⋅3=6 and 1+3=7
Group terms with common factors, factor out common factors from each group, then factor out the resulting common binomial
2x3+3x2−4x−6=x(2x2+3x)−2(2x+3)=(x−2)(2x2+3x+3) groups 2x3+3x2 and −4x−6
Use the quadratic formula when factoring is not possible or efficient
Special polynomial factoring forms
Perfect square trinomials in the form a2+2ab+b2=(a+b)2 or a2−2ab+b2=(a−b)2
x2+6x+9=(x+3)2 is a perfect square trinomial
Difference of squares in the form a2−b2=(a+b)(a−b)
x2−16=(x+4)(x−4) factors as the difference of squares
Factoring cubes expressions
Sum of cubes a3+b3=(a+b)(a2−ab+b2)
x3+8=(x+2)(x2−2x+4) factors as sum of cubes
Difference of cubes a3−b3=(a−b)(a2+ab+b2)
x3−27=(x−3)(x2+3x+9) factors as difference of cubes
Factoring with non-integer exponents
Fractional exponents treated as variables when factoring
4x1/2+6x1/4−2=2(2x1/2+3x1/4−1) factors out common factor of 2
Factor expressions with negative exponents as usual, then simplify the resulting factors
9x−2−4=(3x−1+2)(3x−1−2)=(x3+2)(x3−2) factors then simplifies negative exponents
Advanced factoring techniques
Use the discriminant to determine the nature of a quadratic equation's roots
Apply synthetic division to efficiently divide polynomials by linear factors
Utilize the rational root theorem to find potential rational roots of polynomial equations
Key Terms to Review (2)
Factor by grouping: Factor by grouping is a method used to factor polynomials that involve grouping terms with common factors and then factoring out the greatest common factor from each group. This technique is particularly useful for polynomials with four or more terms.
Greatest common factor: The greatest common factor (GCF) of two or more integers is the largest positive integer that divides each of the numbers without leaving a remainder. It is used to simplify fractions and factor polynomials.