Glossary

6.4 General Strategy for Factoring Polynomials

3 min read•june 24, 2024

Factoring polynomials is a crucial skill in algebra. It's about breaking down complex expressions into simpler parts. This process helps solve equations, simplify fractions, and understand the behavior of functions.

There are several methods for factoring, including finding common factors and special patterns. Mastering these techniques allows you to tackle a wide range of problems. It's like having a toolkit to disassemble mathematical expressions.

Factoring Polynomials

Factoring methods for polynomials

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    • Identify the GCF by finding the highest common factor of all coefficients and the highest power of each variable that appears in every term
    • Divide each term by the GCF and write the result inside parentheses
    • Write the GCF outside the parentheses to complete the factored expression (6x2+9x=3x(2x+3)6x^2 + 9x = 3x(2x + 3))
    • Factoring trinomials of the form ax2+bx+cax^2 + bx + c, where a=1a = 1
      • Multiply the coefficient of x2x^2 (which is 1) and the cc to get the product
      • Find a pair of factors of this product that add up to the coefficient of xx (x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3))
    • Factoring trinomials of the form ax2+bx+cax^2 + bx + c, where a≠1a \neq 1 (also known as a )
      • Multiply the coefficient of x2x^2 (aa) and the constant term (cc) to get acac
      • Find a pair of factors of acac that add up to the coefficient of xx (bb)
      • Write the factored expression using these factors, grouping aa with one factor and cc with the other (2x2+7x+3=(2x+1)(x+3)2x^2 + 7x + 3 = (2x + 1)(x + 3))
    • Group the first two terms and the last two terms of a four-term polynomial
    • Factor out the GCF from each group
    • If the remaining terms in both groups are the same, factor out this common (ax+ay+bx+by=a(x+y)+b(x+y)=(a+b)(x+y)ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y))

Complete factorization techniques

  • Identify the type of polynomial and choose the appropriate factoring method
  • Apply the selected factoring method to break down the polynomial into its factors
  • If any of the resulting factors can be factored further, repeat the process on those factors
  • Continue factoring until all factors are irreducible (cannot be factored further)
  • Write the final answer as a product of (12x2+14x−6=2(6x2+7x−3)=2(3x−1)(2x+3)12x^2 + 14x - 6 = 2(6x^2 + 7x - 3) = 2(3x - 1)(2x + 3))

Special cases in polynomial factoring

    • A trinomial is a perfect square if it has the form a2+2ab+b2a^2 + 2ab + b^2 or a2−2ab+b2a^2 - 2ab + b^2
    • Factor perfect square trinomials by taking the of the first and last terms and using the same sign as the middle term (x2+6x+9=(x+3)2x^2 + 6x + 9 = (x + 3)^2, x2−10x+25=(x−5)2x^2 - 10x + 25 = (x - 5)^2)
    • A polynomial is a difference of cubes if it has the form a3−b3a^3 - b^3
    • Factor the difference of cubes using the formula a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2) (8x3−27=(2x−3)(4x2+6x+9)8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9))
    • A polynomial is a sum of cubes if it has the form a3+b3a^3 + b^3
    • Factor the sum of cubes using the formula a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2) (x3+8=(x+2)(x2−2x+4)x^3 + 8 = (x + 2)(x^2 - 2x + 4))

Understanding Polynomial Factorization

  • A polynomial is an expression consisting of variables and coefficients, using only addition, subtraction, and multiplication operations
  • is the process of breaking down a polynomial into simpler expressions that, when multiplied together, produce the original polynomial
  • A binomial is a polynomial with two terms, often used in factoring techniques
  • Irreducible factors are expressions that cannot be factored further using real numbers

Key Terms to Review (17)

±: The symbol '±' is a mathematical symbol that represents the concept of 'plus or minus'. It is used to indicate that a value or expression can have two possible solutions, one positive and one negative. This symbol is commonly encountered in the context of solving equations and working with polynomial expressions.
Ac Method: The ac method is a specific technique used in the factoring of polynomials, particularly trinomials. It involves identifying the constant term (c) and the coefficient of the squared term (a) in order to determine the factors of the polynomial.
Binomial: A binomial is a polynomial expression that consists of two terms, typically connected by addition or subtraction operations. It is a fundamental concept in algebra that is essential for understanding and manipulating polynomial expressions.
Complete Factorization: Complete factorization is the process of breaking down a polynomial expression into its prime factors. This technique is a crucial step in the general strategy for factoring polynomials, as it allows for the identification of all the factors that contribute to the structure of the expression.
Constant Term: The constant term is a numerical value that does not have a variable associated with it in a polynomial expression. It is the term that remains unchanged regardless of the value assigned to the variable(s) in the expression.
Difference of Cubes: The difference of cubes is a special product in algebra where the difference between two cubes (the third power of a number) can be factored into a simpler expression. This concept is important in understanding how to multiply and factor polynomials, as well as solve polynomial equations.
Factoring by Grouping: Factoring by grouping is a technique used to factor polynomials by first grouping the terms in the polynomial, then identifying a common factor within each group, and finally factoring out the greatest common factor (GCF) to simplify the expression. This method is particularly useful for polynomials that do not have a clear common factor among all the terms.
Factoring Out the Greatest Common Factor (GCF): Factoring out the greatest common factor (GCF) is a fundamental technique in polynomial factorization. It involves identifying the largest factor that is common to all the terms in a polynomial expression and extracting it as a common factor, which simplifies the expression and makes it easier to factor further.
Factoring Trinomials: Factoring trinomials is the process of expressing a polynomial with three terms as a product of two binomials. This technique is a crucial component of the general strategy for factoring polynomials, as it allows for the simplification and manipulation of more complex algebraic expressions.
Factorization: Factorization is the process of breaking down a polynomial or an expression into a product of smaller, simpler factors. It involves identifying the common factors among the terms and expressing the original expression as a product of these factors. Factorization is a fundamental technique in algebra that is essential for solving polynomial equations and simplifying algebraic expressions.
Irreducible Factors: Irreducible factors are polynomial factors that cannot be further factored into simpler polynomial expressions. They represent the building blocks of more complex polynomial equations and are a crucial concept in the general strategy for factoring polynomials.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It represents the scale or magnitude of the polynomial and plays a crucial role in various polynomial operations and properties.
Perfect Square Trinomials: A perfect square trinomial is a polynomial expression of the form $a^2 + 2ab + b^2$, where $a$ and $b$ are real numbers. These trinomials are called 'perfect' because they can be factored into the square of a binomial expression, $(a + b)^2$.
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 algebra and play a crucial role in various mathematical topics covered in this course.
Quadratic Expression: A quadratic expression is a polynomial expression of degree two, containing a variable with the highest exponent being two. These expressions are characterized by their ability to be factored or solved using various algebraic techniques, making them an essential component in the study of algebra.
Square Root: The square root, denoted by the symbol √, is a mathematical operation that represents the inverse of squaring. It is used to find the value that, when multiplied by itself, results in the original number. The square root of a number is the value that, when raised to the power of 2, equals the original number.
Sum of Cubes: The sum of cubes is a mathematical expression that represents the sum of two or more numbers raised to the third power. This concept is particularly relevant in the context of multiplying polynomials, factoring special products, and solving polynomial equations.
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