Problem 36 - Technical Mathematics with Calculus

Question

Factor completely, by hand or by calculator. Check your results. The General Quadratic Trinomial. $$9 x^{2}-27 x+18$$

Step-by-Step Solution

Answer

The factored form of the quadratic trinomial \(9 x^{2}-27 x+18\) is \(3(x - 1)(3x - 6)\).

Step 1: Identify the Trinomial

Recognize the expression as a quadratic trinomial in the form of \(ax^2 + bx + c\), where \(a = 9\), \(b = -27\), and \(c = 18\).

Step 2: Setup for Factoring

To factor the trinomial, we are looking for two binomials in the form of \((dx + e)(fx + g)\), where the product of \(d\) and \(f\) equals \(a\), and the product of \(e\) and \(g\) equals \(c\), and the sum of \(de\) and \(fg\) equals \(b\).

Step 3: Finding Numbers to Factor

Determine two numbers that multiply to \(a \cdot c = 9 \cdot 18 = 162\) and add up to \(b = -27\). The numbers that fit this criteria are \(-9\) and \(-18\).

Step 4: Rewrite the Trinomial

Rewrite the middle term using the two numbers found in step 3 to express the trinomial as a four-term polynomial: \(9x^2 - 9x - 18x + 18\).

Step 5: Factor by Grouping

Group the terms to factor by grouping: \((9x^2 - 9x) + (-18x + 18)\).

Step 6: Factor Out the Common Factor in Each Group

Factor out the greatest common factor (GCF) from each group: \(9x(x - 1) - 18(x - 1)\).

Step 7: Factor Out the Common Binomial Factor

Factor out the common binomial factor \((x - 1)\), which gives us \((x - 1)(9x - 18)\).

Step 8: Final Factoring

The second binomial can be factored further since 9 is a common factor. The fully factored form is: \((x - 1)(9x - 18) = (x - 1)(3(3x - 6)) = (x - 1)(3(3x - 6)) = 3(x - 1)(3x - 6)\).

Step 9: Check the Result

To check the factoring, expand the factored form to see if it matches the original polynomial. \((x - 1)(3x - 6)\) expands to \(3(x(3x) - x(6) - 1(3x) + 1(6))\) which simplifies to \(9x^2 - 27x + 18\).

Key Concepts

Quadratic Trinomial

When discussing polynomials, a quadratic trinomial stands out as an essential concept. It is a second-degree polynomial, meaning it has a power of two as its highest degree, and it's characterized by three terms. These terms usually take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \eq 0 \).

To successfully factor quadratic trinomials, understanding the relationship between these constants is key. We want to break down the trinomial into the product of two binomials, each containing a variable term \( x \) and a constant term. This process gives us insight into the roots of the equation since the values that \( x \) can take to render the polynomial equivalent to zero originate from these binomials. For example, in the trinomial \( 9x^2 - 27x + 18 \), it's these constants that guide the factoring process.

Factor by Grouping

The strategy of factoring by grouping is particularly useful when we are dealing with polynomials that can be arranged into groups with a common factor. Initially, it might seem odd to introduce extra terms to simplify an expression, but this method provides a pathway to simplification.

Once the terms of the polynomial are organized into groups, we can then extract the greatest common factor (GCF) from each group. The goal here is to reveal a common binomial factor from the separated groups, which then unites them into a simpler, factored form of the original polynomial.

Finding the GCF in Each Group

In our case, grouping the terms from the modified trinomial \( 9x^2 - 9x - 18x + 18 \) reveals two pairs: the first contains \( 9x \) as a GCF, and the second \( -18 \). Factoring out the GCFs presents us with a shared binomial that is factored out, simplifying the expression to \( (x - 1)(9x - 18) \).

Greatest Common Factor (GCF)

Determining the greatest common factor (GCF) is an important step in polynomial factoring. The GCF is the highest number that divides exactly into two or more numbers. Identifying and extracting the GCF from a polynomial's terms often simplifies the expression and is integral in factoring by grouping.

In our example, we seek the GCF in each of the groups within \( 9x^2 -9x \) and \( -18x + 18 \). For the first group, \( 9x \) is the GCF, and for the second, the GCF is \( -18 \). We pull these factors out to reveal a common factor of \( (x - 1) \), which paves the way for a more straightforward factored expression.

Polynomial Factoring Steps

The process of polynomial factoring consists of a series of steps designed to simplify complex expressions into products of smaller, more manageable polynomials. To begin, recognize the type of polynomial you are working with and then apply the appropriate method, like factoring by grouping for four-term polynomials, or direct factoring for trinomials.

The

Steps of Factoring

generally involve finding the GCF if one exists, examining the structure of the polynomial to determine a factoring strategy, and then systematically applying factoring techniques. In complicated cases, you may need to manipulate the terms (by splitting the middle term, for instance) to facilitate the factoring process. Once factored, the solution should always be verified by expanding the factored form and checking if the original polynomial is attained. As seen in the exercise, the factoring steps culminate in the expression \( 3(x - 1)(3x - 6) \) that simplifies from the original trinomial \( 9x^2 - 27x + 18 \).