Problem 15 - Introductory Algebra for College Students

Question

Before getting to multiple-step factorizations, let's be sure that you are comfortable with exercises requiring only one of the factoring techniques. Factor each polynomial. $$6 x^{2}+x-15$$

Step-by-Step Solution

Answer

The factorized form of the trinomial \(6x^{2}+x-15\) is \((2x -3)(3x + 5)\)

Step 1: Identify and write down the trinomial

The given trinomial is \(6x^2 + x - 15\).

Step 2: Determine the coefficient and constant terms

The coefficient of the quadratic term (x^2) is 6 and the constant term is -15.

Step 3: Find the pair of numbers

Find two numbers such that they multiply to give -90 (i.e. 6 times -15) and they add up to 1 (the coefficient of x). In this case, the pair of numbers that satisfy these conditions should be 10 and -9.

Step 4: Rewrite the middle term

Split the middle term, \(x\), of the trinomial using the pair of numbers 10 and -9. This step results in the equation \(6x^2 + 10x - 9x - 15\).

Step 5: Factor by grouping

Group the terms as \((6x^2 + 10x) - (9x + 15)\). Now, factor out common terms from each group. This results in \(2x(3x + 5) - 3(3x + 5)\).

Step 6: Simplify

Finally, note that \((3x + 5)\) is a common factor in both terms. Therefore, the factorized form of the trinomial is \((2x -3)(3x + 5)\).

Key Concepts

Trinomial

A trinomial is a special type of polynomial that consists of exactly three terms. When working with trinomials like \(6x^2 + x - 15\), you'll always have a quadratic term, a linear term, and a constant term.
In this example:

• The quadratic term is \(6x^2\).
• The linear term is \(x\).
• The constant term is \(-15\).

Each trinomial can potentially be factorized into two binomials, which is a useful technique to simplify expressions and solve equations.
Being able to identify the terms in a trinomial is crucial as it sets you up for the next steps in factorization. You start by finding numbers that work both in multiplication for the quadratic and constant terms, and addition or subtraction for the linear term.

Factoring by Grouping

Factoring by grouping is a handy trick to tackle tricky polynomials like a trinomial. Once you've got your trinomial, break it apart to make the process easier.
For instance, with the trinomial \(6x^2 + x - 15\), after identifying pairs like 10 and -9 that work with the coefficients, rewrite it as \(6x^2 + 10x - 9x - 15\).
Next up, group the terms: \((6x^2 + 10x) - (9x + 15)\). Why group? Well, it helps you simplify by factoring out common factors from each little group! Here's how:

• In \(6x^2 + 10x\), \(2x\) is common, giving you \(2x(3x + 5)\).
• In \(-9x - 15\), \(-3\) is common, simplifying to \(-3(3x + 5)\).

Both groups include \((3x + 5)\), so you extract this as a common factor for the final simplification.

Polynomial Coefficients

Coefficients are the numerical part of a term in a polynomial that tells you how many of that variable you have. In \(6x^2 + x - 15\):

• 6 is the coefficient of \(x^2\), meaning you have six \(x^2\) terms.
• 1 is implied as the coefficient of \(x\) here, so there is just one \(x\) term.
• -15 is the constant term and doesn't have a variable attached.

Understanding coefficients is essential because these numbers affect how you group terms and find factor pairs.
When factorizing, consider both the interactions between the coefficients in terms of multiplication and addition, as they determine the pairs of numbers you need to use to successfully split and factor the polynomial.
Knowing how to manipulate these coefficients is vital when aiming to simplify polynomials into a factorized form.