Factoring Trinomials & Polynomials, Basic Introduction - Algebra

This paragraph introduces the concept of factoring by finding the greatest common factor (GCF). It explains how to factor expressions such as 3x + 15 and 7x - 28 by dividing each term by their GCF, which are 3 and 7 respectively. The paragraph also presents additional examples including 4x^2 + 8x and 5x^2 - 15x^3, illustrating the process of factoring by removing the GCF and simplifying the expressions. The importance of checking work by multiplying the factors is emphasized.

The second paragraph delves into the method of factoring by grouping, a technique used for polynomial expressions with four terms. It uses the example of x^3 - 4x^2 + 3x - 12 to demonstrate how to separate the expression into two groups, factor out the GCF from each group, and then combine like terms to rewrite the expression in factored form. Another example, 2r^3 - 6r^2 + 5r - 15, is provided for practice, highlighting the importance of identifying a common term to successfully factor by grouping.

This section focuses on factoring trinomials where the leading coefficient is 1. It provides a step-by-step guide on finding two numbers that multiply to the constant term and add up to the linear coefficient. Examples include factoring expressions like x^2 + 7x + 12 into (x + 3)(x + 4) and x^2 + 3x - 20 into (x + 7)(x - 4). The process involves listing factors of the constant term and identifying pairs that sum to the linear coefficient.

The paragraph discusses how to factor trinomials when the leading coefficient is not 1. It suggests multiplying the leading coefficient by the constant term and finding two numbers that multiply to this product and add up to the linear coefficient. An example is given with the expression 2x^2 - 9x + 20, which is factored by first factoring out the GCF of 2 and then using the technique described to factor the remaining quadratic expression.

This part of the script covers the factoring of perfect square trinomials and the difference of squares. It explains that a perfect square trinomial is of the form a^2 + 2ab + b^2 and can be factored into (a + b)^2. The difference of squares, a^2 - b^2, can be factored into (a + b)(a - b). Examples are provided to illustrate the identification of perfect square trinomials and the application of the respective formulas for factoring.

The script explains the factoring of sums and differences of cubes, providing the formulas for each: sum of cubes a^3 + b^3 factors into (a + b)(a^2 - ab + b^2), and difference of cubes a^3 - b^3 factors into (a - b)(a^2 + ab + b^2). It includes examples demonstrating the process of identifying cubes and applying the formulas to factor expressions like x^3 + 8 and 8x^3 - 27.

The final paragraph of the script discusses solving equations by factoring. It starts with a simple example of a binomial equation 6x^2 - 30x = 0, showing how to factor out the GCF and use the zero product property to find the values of x. Further examples are given, including solving 3x^2 - 27 and x^2 - 5x - 36 by factoring and applying the zero product property to determine the solutions for x.