Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Factor the trinomial.\(9 x^{2}-3 x-2\)

Step-by-Step Solution

Answer

The factorized form of \(9x^{2}-3x-2\) is \((3x-2)(3x+1)\)

Step 1: Identify a, b, c

From the standard form \(ax^2 + bx + c\) of the quadratic equation, set \(a=9\), \(b=-3\), and \(c=-2\).

Step 2: Determine the Factors

Determine two numbers that multiply to \(ac=-18\) and add up to \(b=-3\). From inspection, these numbers are -6 and 3. Since -6 multiplied by 3 equals -18 and -6 added to 3 equals -3.

Step 3: Rewrite Middle Term

Split the middle term using those two numbers and rewrite the equation: \(9x^{2} - 6x + 3x -2\)

Step 4: Factor By Grouping

Factorise the polynomial by grouping the first two terms together and the last two terms together. Here, we get \(3x(3x-2) + 1(3x-2)\)

Step 5: Combine Like Terms

Combine the two terms together to get the factored form of the trinomial. We see that both terms have a common binomial factor of \(3x-2\). Thus, the trinomial in factored form becomes \((3x-2)(3x+1)\)

Step 6: Check By Expanding

The final step is to check the factored equation by expanding it to verify that it gives the original equation. Expand by performing the distributive property twice to get an expression equivalent to the original trinomial.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra and appear frequently in various mathematical problems. At their core, they are polynomial equations of degree two, often expressed in the form: \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants where \(a eq 0\). They are called quadratic because of the "quad" root in the word, which means square.
Quadratic equations graph as parabolas, which can open upwards or downwards depending on the sign of \(a\). For students, solving them involves both factoring techniques and understanding their graphical representations.
Key characteristics of quadratic equations include:

• Vertex: The highest or lowest point of the parabola.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
• Roots: The x-values where the equation equals zero. Finding these is a common task when solving the equations.

When factoring trinomials like \(9x^2 - 3x - 2\), identifying the roots can verify if the factorization is correct, as these roots can also be calculated by setting the factors equal to zero and solving for \(x\). This step ensures that the factorization process is accurate.

Polynomial Factoring

Polynomial factoring is an essential skill in algebra that simplifies complex expressions. In the context of quadratic equations, it involves breaking down a trinomial into simpler binomial factors. This process is crucial for solving equations, simplifying expressions, and finding zeroes of functions. It allows students to convert complex trinomials into products of simpler expressions.
The primary goal of factoring a polynomial like \(9x^2 - 3x - 2\) is to express it as the product of two polynomials. Factoring involves steps such as:

• Identify the coefficients: From the equation \(ax^2 + bx + c\), determine \(a\), \(b\), and \(c\).
• Find appropriate factors: Search for two numbers that fit the criteria \(ac\) and \(b\).
• Apply grouping technique: Split the middle term using these factors for easier grouping.

Knowing how to factor is incredibly helpful because it simplifies solving quadratic equations by making them easier to interpret or graph. It is an algebra technique that serves as a precursor to more advanced methods like completing the square or using the quadratic formula.

Algebra Techniques

Algebra techniques are the toolbox tools necessary to manipulate, simplify, and solve mathematical equations. When working with quadratic equations and factoring, several key techniques come into play. For example, the process of factoring by grouping as seen in the trinomial \(9x^2 - 3x - 2\) involves breaking terms into compatible pairs that simplify further through factoring.
Effective algebra techniques include:

• Distributive Property: Used in both expanding and simplifying expressions.
• Grouping: A technique to simplify an expression by organizing terms into pairs or groups.
• Common Factor Extraction: Identifying and simplifying expressions by taking out common factors.

In the context of the provided solution, using these techniques allows for the breaking down of the polynomial into manageable parts. By approaching the polynomial with these methods, students can better understand the structure and solution of algebraic expressions, building a solid foundation for more intricate problems later.