Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 12 x^{2}-x-6 $$

Step-by-Step Solution

Answer

The factored form is \((4x - 3)(3x + 2)\).

Step 1: Identify Coefficients

The given polynomial is \( 12x^2 - x - 6 \). Identify the coefficients: \( a = 12 \), \( b = -1 \), and \( c = -6 \).

Step 2: Multiply 'a' and 'c'

Multiply the coefficient of \( x^2 \) (\( a = 12 \)) by the constant term (\( c = -6 \)): \( 12 imes -6 = -72 \).

Step 3: Find Two Numbers

Find two numbers that multiply to \(-72\) and add to the middle coefficient \(-1\). The numbers are \(8\) and \(-9\).

Step 4: Rewrite the Middle Term

Rewrite the polynomial by splitting the \(-x\) term into \(8x - 9x\): \( 12x^2 + 8x - 9x - 6 \).

Step 5: Factor by Grouping

Group the terms: \((12x^2 + 8x) + (-9x - 6)\). Factor each group: \(4x(3x + 2) - 3(3x + 2)\).

Step 6: Factor Out Common Binomial

Notice that \(3x + 2\) is common in both groups, factor it out: \((4x - 3)(3x + 2)\).

Step 7: Verify the Factored Form

To ensure the factorization is correct, expand \((4x - 3)(3x + 2)\) and verify it equals the original polynomial: \(12x^2 + 8x - 9x - 6 = 12x^2 - x - 6\).

Key Concepts

Quadratic Polynomials

Quadratic polynomials are a type of polynomial that are extremely common in algebra. They take the general form: \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.
Here is what each part means:

• \( a \) is the coefficient of the quadratic term \( x^2 \).
• \( b \) is the coefficient of the linear term \( x \).
• \( c \) is the constant term with no variable attached.

Quadratic polynomials can be solved in many different ways, including factoring, using the quadratic formula, or completing the square. Each method provides a way to find the values of \( x \) that satisfy the quadratic equation when set to zero, which are also known as the roots or solutions of the polynomial.
In our example, the quadratic polynomial is \( 12x^2 - x - 6 \), where \( a = 12 \), \( b = -1 \), and \( c = -6 \). The goal is to factor this expression into a product of simpler polynomials.

Factoring by Grouping

Factoring by grouping is a method used to factor certain polynomial expressions. It involves rearranging the terms of a polynomial and grouping them in pairs in such a way that each pair can be factored separately.
Here's a step-by-step breakdown of how it works:

• First, identify any common factors in the groups of terms. These groups are usually selected so that a common factor exists.
• Then, factor out the common factor from each group.
• Finally, if there is a common binomial factor between the two groups, factor it out of the expression.

In our case, the polynomial \( 12x^2 - x - 6 \) was rewritten as \( 12x^2 + 8x - 9x - 6 \). The terms were grouped as \((12x^2 + 8x) + (-9x - 6)\). From each group, \( 4x \) and \(-3\) are factored out respectively, resulting in \( 4x(3x + 2) - 3(3x + 2) \). Notice how \( 3x + 2 \) appears in both terms - this is the common binomial factor that can be factored out, leading to the final factorized form \((4x - 3)(3x + 2)\).

Polynomial Coefficients

Polynomial coefficients are the numbers that multiply the variable terms in a polynomial. They are crucial in determining the characteristics and behavior of the polynomial.
Let's explore this concept further:

• The leading coefficient is the number in front of the highest degree term, determining the polynomial's global shape.
• The constant term, or the term with no variable factor, provides the polynomial's value when all variables are zero.
• Intermediate coefficients affect the polynomial's direction and symmetry.

For the polynomial \( 12x^2 - x - 6 \):

• The leading coefficient \( a \) is 12, indicating that as \( x \) becomes very large or very small, \( \pm12x^2 \) will dominate the polynomial's behavior.
• The linear coefficient \( b \) is -1, influencing the slope of the parabola represented by the polynomial when graphing.
• The constant term \( c \) is -6, which is the value of the polynomial when \( x = 0 \).

Understanding coefficients helps us factor, graph, and solve polynomials effectively.