Problem 28 - College Algebra

Question

In Exercises \(17-30,\) factor each trinomial, or state that the trinomial is prime. $$6 x^{2}-17 x+12$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(6x^{2}-17x+12\) is \((2x-3)(3x-4)\).

Step 1: Identify Multiplication Product

Identify the two numbers that multiply up to 72 (6*12) and at the same time, they add up to -17.

Step 2: Find the two numbers

The two numbers that satisfy these conditions are -8 and -9. Because -8 multiplied by -9 equals 72; and -8 plus -9 equals -17.

Step 3: Split the middle term

We split the middle term of the trinomial into two terms using the two new found numbers. The trinomial thus becomes \(6x^{2}-8x -9x+12\).

Step 4: Regroup and Factor by Grouping

Next, group the first two terms together and the last two terms together, it becomes \((6x^{2}-8x) - (9x-12)\). Factoring out from each group we get \(2x(3x-4)-3(3x-4)\).

Step 5: Factoring the result

Now the expression \(2x(3x-4)-3(3x-4)\) has a common factor of \((3x-4)\). Factor out this common factor to get the final answer.

Key Concepts

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give the original polynomial. In algebra, this is a core skill, crucial for solving equations and simplifying expressions.

For example, factoring the trinomial 6x^2-17x+12 begins with identifying a pair of numbers that both add up to the coefficient of the x term (in this case, -17) and multiply to the product of the coefficient of the x^2 term and the constant term (here, 6 * 12 = 72). Discovering that -8 and -9 are the numbers that satisfy these conditions, the trinomial can then be factored as follows:
6x^2 - 17x + 12 = (3x-4)(2x-3).

The ability to factor polynomials is essential not just for solving quadratic equations but also for analyzing graphs of functions, finding zeros, and integrating algebraic functions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra and allow us to describe patterns, relationships, and changes. A trinomial is a specific type of algebraic expression that consists of three terms.

For instance, 6x^2-17x+12 is a trinomial where 6x^2 is the quadratic term, -17x is the linear term, and 12 is the constant term. Each term comprises a coefficient and a variable raised to an exponent. Understanding how to manipulate these expressions, such as through factorization, enables students to solve a broad range of algebraic problems.

Grouping Method in Algebra

The grouping method is a technique used in algebra to factor polynomials that do not easily factor into a product of binomials. The method involves rearranging and grouping terms in such a way that they share a common factor, which can then be factored out.

In the case of the trinomial 6x^2-17x+12, after finding numbers that multiply to 72 and add to -17, we rewrite the middle term as two terms: -8x and -9x. This step allows us to regroup the trinomial into two binomials: (6x^2-8x) and (-9x+12). From each pair, we factor out the greatest common factor, giving us 2x(3x-4) and -3(3x-4). Since both groups contain a (3x-4), we can factor this out to get the final factored form of the trinomial: (3x-4)(2x-3).

This method not only simplifies complex polynomials but also is an excellent strategy for students to check their work for possible common factors in polynomial expressions.