1.5 Factoring Polynomials
4 min read•june 24, 2024
Factoring polynomials is a key skill in algebra, breaking down complex expressions into simpler parts. It's like solving a puzzle, finding the pieces that multiply together to form the original . This process helps simplify equations and solve problems more easily.
From greatest common factors to special patterns like , there are various techniques to factor polynomials. Mastering these methods opens doors to solving more advanced math problems and understanding algebraic relationships better.
Factoring Polynomials
Introduction to Polynomials
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- A is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication
- Polynomials can be classified based on the number of terms: • : a polynomial with one term • : a polynomial with two terms • : a polynomial with three terms
- is the process of breaking down a polynomial into the product of simpler expressions
Greatest common factor in polynomials
- Find the largest factor that divides all terms in the polynomial without leaving a remainder
- Factor each term completely to identify common factors among all terms
- Select the factor with the highest power for each variable and the largest numerical as the GCF (12, , )
- Divide each term by the GCF and place the GCF outside the parentheses to factor out the GCF ()
Methods for quadratic trinomials
- Quadratic trinomials have the form [ax^2 + bx + c](https://www.fiveableKeyTerm:ax^2_+_bx_+_c), where , , and are constants and ()
- Factoring by grouping involves grouping the first two terms and the last two terms, from each group, and factoring out the common binomial if the remaining terms in the parentheses are the same ()
- multiplies and to get , finds two numbers that multiply to give and add to give , rewrites the middle term using these numbers, and factors by grouping ()
- finds the roots by substituting , , and , and the factored form is written using the roots ()
Grouping method for longer polynomials
- Group terms into pairs with a common factor, factor out the GCF from each group, and factor out the common binomial if the remaining terms are the same ()
- Repeat the process if necessary until the polynomial is fully factored ()
Perfect square trinomials
- Perfect square trinomials have the form or (, )
- Take the square root of the first and last terms, determine the sign based on the middle term, and write the factored form as (, )
Difference of squares
- has the form [a^2 - b^2](https://www.fiveableKeyTerm:a^2_-_b^2) (, )
- Take the square root of each term and write the factored form as (, )
Sum and difference of cubes
- has the form [a^3 + b^3](https://www.fiveableKeyTerm:a^3_+_b^3), and has the form [a^3 - b^3](https://www.fiveableKeyTerm:a^3_-_b^3) (, )
- For sum of cubes, write the factored form as ()
- For difference of cubes, write the factored form as ()
Factoring with non-integer exponents
- For fractional exponents, rewrite the expression using properties of exponents to eliminate fractional exponents and factor the resulting polynomial ()
- For negative exponents, rewrite the expression using properties of exponents to eliminate negative exponents and factor the resulting polynomial ()
Key Terms to Review (32)
A^2 - b^2: The expression a^2 - b^2 represents the difference between the squares of two variables, a and b. This expression is a special case of a polynomial subtraction and is a fundamental concept in factoring polynomials.
A^3 - b^3: The expression $a^3 - b^3$ represents the difference between the cubes of two variables, $a$ and $b$. This term is particularly important in the context of factoring polynomials, as it can be factored using a specific method known as the difference of cubes.
A^3 + b^3: The expression $a^3 + b^3$ represents the sum of the cubes of two variables, $a$ and $b$. This expression is particularly important in the context of factoring polynomials, as it can be factored using a special factorization method known as the sum of cubes formula.
AC Method: The AC method, also known as the Associative and Commutative method, is a technique used in the context of factoring polynomials. It allows for the efficient factorization of polynomial expressions by taking advantage of the associative and commutative properties of multiplication.
Ax^2 + bx + c: The general form of a quadratic equation, where 'a', 'b', and 'c' are coefficients that represent the parameters of the equation. This expression is central to the process of factoring polynomials, as it provides the foundation for identifying and manipulating the factors that make up the quadratic expression.
Binomial: A binomial is a polynomial with exactly two terms. It is a mathematical expression that consists of two monomials, which are variables or constants combined by addition or subtraction.
Binomial coefficient: A binomial coefficient is a coefficient of any of the terms in the expansion of a binomial raised to a power, typically written as $\binom{n}{k}$ or $C(n,k)$. It represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the magnitude or strength of the relationship between the variable and the overall expression. Coefficients are essential in various mathematical contexts, including polynomial factorization, linear equations, quadratic equations, and the graphing of polynomial functions.
Difference of Cubes: The difference of cubes is a special case of polynomial factorization where a polynomial expression can be factored by recognizing the difference between two cubes. This factorization technique is useful in solving certain types of equations and understanding the behavior of power functions and polynomial functions.
Difference of squares: The difference of squares is a specific type of polynomial that takes the form $a^2 - b^2$, which can be factored into $(a + b)(a - b)$. It is based on the property that the product of a sum and a difference of two terms results in the difference of their squares.
Difference of Squares: The difference of squares is a special type of polynomial expression where the terms are the difference between two perfect squares. This concept is particularly important in the context of factoring polynomials, working with rational expressions, solving quadratic equations, and understanding the properties of power functions and polynomial functions.
Factor by Grouping: Factoring by grouping is a technique used to factor polynomials by identifying common factors among groups of terms within the polynomial. This method involves dividing the polynomial into smaller groups, finding the common factor of each group, and then factoring out that common factor to simplify the expression.
Factoring out the GCF: Factoring out the Greatest Common Factor (GCF) is a technique used in algebra to simplify polynomial expressions by identifying and extracting the largest common factor shared by all the terms in the expression. This process helps to break down the polynomial into smaller, more manageable parts, making it easier to work with and understand.
Factoring Strategies: Factoring strategies are mathematical techniques used to break down polynomials into their prime factors. These strategies help simplify algebraic expressions and solve equations by identifying common factors and patterns within the polynomial.
Factorization: Factorization is the process of expressing a polynomial or an algebraic expression as a product of smaller, simpler expressions. It involves breaking down a complex expression into a multiplication of two or more factors that, when multiplied together, result in the original expression.
FOIL Method: The FOIL method is a systematic approach used to multiply two binomials, or expressions with two terms. The acronym FOIL stands for the order in which the terms are multiplied: First, Outer, Inner, Last.
Greatest common factor: The greatest common factor (GCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. It is useful in simplifying fractions, factoring polynomials, and solving equations.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides two or more integers without a remainder. It is a fundamental concept in algebra that is essential for understanding real numbers, factoring polynomials, working with rational expressions, solving quadratic equations, and analyzing power and polynomial functions.
Greatest Common Factor (GCF): The Greatest Common Factor (GCF) is the largest positive integer that divides each of the given integers or polynomial terms without a remainder. It is a fundamental concept in factoring polynomials, as identifying the GCF allows for efficient factorization and simplification of algebraic expressions.
Monomial: A monomial is a polynomial with only one term, which can be a constant, a variable, or a product of constants and variables raised to non-negative integer powers.
Monomial: A monomial is a single algebraic term that consists of a numerical coefficient and one or more variables raised to non-negative integer powers. It is the fundamental building block of polynomials, which are expressions formed by the sum of monomials.
Perfect square trinomial: A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$.
Perfect Square Trinomial: A perfect square trinomial is a special type of polynomial expression that can be factored as the square of a binomial. It is a three-term polynomial in the form $a^2 + 2ab + b^2$, where $a$ and $b$ are real numbers.
Polynomial: A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents. It can be written in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ where $a_n, a_{n-1}, ..., a_1, a_0$ are constants and $n$ is a non-negative integer.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers. Polynomials are fundamental in various areas of mathematics, including algebra, calculus, and the study of functions.
Polynomial Factorization: Polynomial factorization is the process of breaking down a polynomial expression into a product of simpler polynomial factors. This technique is essential in the study of 1.5 Factoring Polynomials, as it allows for the simplification and manipulation of complex polynomial expressions.
Quadratic formula: The quadratic formula is used to find the roots of a quadratic equation of the form $ax^2 + bx + c = 0$. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. This formula provides a systematic way to find the solutions, or roots, of a quadratic equation.
Quadratic trinomial: A quadratic trinomial is a polynomial of degree two that can be expressed in the standard form of $$ax^2 + bx + c$$, where 'a', 'b', and 'c' are constants and 'a' is not zero. This expression consists of three terms: a quadratic term ($$ax^2$$), a linear term ($$bx$$), and a constant term ($$c$$). Quadratic trinomials are significant in factoring polynomials as they can often be factored into the product of two binomials.
Sum of Cubes: The sum of cubes is a mathematical expression that represents the sum of the cubes of two or more numbers. It is an important concept in various areas of mathematics, including factoring polynomials, solving certain types of equations, and understanding the behavior of power functions and polynomial functions.
Trinomial: A trinomial is a polynomial expression with three terms, typically in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are coefficients and $x$ is a variable. Trinomials are an important concept in the context of factoring polynomials, as they can often be factored into the product of two binomials.
Zero Product Property: The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This property is fundamental in the context of factoring polynomials and solving quadratic equations.