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How to factor a polynomial?

Published in Factoring Polynomials 3 mins read

Factoring a polynomial involves breaking it down into simpler expressions (factors) that, when multiplied together, give you the original polynomial. There are several techniques for factoring polynomials, and the best approach depends on the specific polynomial you are trying to factor. Let's explore some common methods:

Common Factoring Techniques

Here's a breakdown of common factoring techniques:

1. Greatest Common Factor (GCF)

  • What it is: Identifying the largest factor that divides all terms of the polynomial.

  • How to do it:

    1. Find the GCF of the coefficients (numerical part) of the terms.
    2. Find the GCF of the variable parts of the terms (look for the lowest exponent of each variable).
    3. Factor out the GCF from each term of the polynomial.

  • Example (Based on provided reference):

    • Factor 3x³ - 9x²

    • GCF of 3 and 9 is 3.

    • GCF of x³ and x² is x².

    • Therefore, GCF of the polynomial is 3x².

    • Factoring out 3x²: 3x²(x - 3).

    • Factor 4x² - 12x

    • GCF of 4 and 12 is 4.

    • GCF of x² and x is x.

    • Therefore, GCF of the polynomial is 4x.

    • Factoring out 4x: 4x(x - 3).

2. Difference of Squares

  • What it is: Factoring a binomial in the form a² - b².

  • Formula: a² - b² = (a + b)(a - b)

  • Example:

    • Factor x² - 4
    • Recognize that x² is x² and 4 is 2².
    • Apply the formula: (x + 2)(x - 2)

3. Factoring Trinomials (Quadratic Expressions)

  • What it is: Factoring a trinomial in the form ax² + bx + c.

  • Method: Find two numbers that:

    • Multiply to ac (the product of the leading coefficient and the constant term).
    • Add up to b (the coefficient of the middle term).

  • Example:

    • Factor x² + 5x + 6
    • We need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.
    • Therefore, the factored form is (x + 2)(x + 3)

4. Factoring by Grouping

  • What it is: A technique used for polynomials with four or more terms.

  • How to do it:

    1. Group the terms into pairs.
    2. Factor out the GCF from each pair.
    3. If the remaining binomial factors are the same, factor them out.

  • Example:

    • Factor x³ + 2x² + 3x + 6
    • Group the terms: (x³ + 2x²) + (3x + 6)
    • Factor out the GCF from each group: x²(x + 2) + 3(x + 2)
    • Factor out the common binomial (x + 2): (x + 2)(x² + 3)

5. Sum and Difference of Cubes

  • Sum of Cubes Formula: a³ + b³ = (a + b)(a² - ab + b²)

  • Difference of Cubes Formula: a³ - b³ = (a - b)(a² + ab + b²)

  • Example (Sum of Cubes):

    • Factor x³ + 8 (which is x³ + 2³)
    • Apply the formula: (x + 2)(x² - 2x + 4)

Tips for Factoring

  • Always look for a GCF first.
  • Pay attention to the signs in the polynomial.
  • Practice, practice, practice! The more you factor, the better you'll become at recognizing patterns.
  • Don't be afraid to try different methods.
  • Check your answer by multiplying the factors back together to make sure you get the original polynomial.

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