Problem 80 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{6 x^{2}-7 x-5}{2 x^{2}+5 x+2} $$

Step-by-Step Solution

Answer

Simplified expression: \(\frac{3x - 5}{x + 2}\).

Step 1: Factor the Numerator

Let's look at the numerator, \(6x^2 - 7x - 5\). We will factor it by finding two numbers that multiply to \((6)(-5) = -30\) and add to \(-7\). These numbers are \(-10\) and \(3\). So, we rewrite the middle term using these numbers: \[6x^2 - 7x - 5 = 6x^2 - 10x + 3x - 5.\] Next, factor by grouping:\[=(6x^2 - 10x) + (3x - 5),\] factoring out common terms:\[= 2x(3x - 5) + 1(3x - 5),\]and the final factorization is:\[= (2x + 1)(3x - 5).\]

Step 2: Factor the Denominator

Now, let's factor the denominator, \(2x^2 + 5x + 2\). We'll use the same technique, finding two numbers that multiply to \((2)(2) = 4\) and add to \(5\). These numbers are \(4\) and \(1\). So, rewrite the expression:\[2x^2 + 5x + 2 = 2x^2 + 4x + x + 2.\]Factor by grouping:\[=(2x^2 + 4x) + (x + 2),\] factoring out the greatest common factors gives:\[= 2x(x + 2) + 1(x + 2),\]and the final factorization is:\[= (2x + 1)(x + 2).\]

Step 3: Simplify the Expression

We now have the factored form of the fraction:\[\frac{(2x + 1)(3x - 5)}{(2x + 1)(x + 2)}.\]Since \((2x + 1)\) is a common factor in both the numerator and the denominator, we can cancel it out, which simplifies the expression to:\[\frac{3x - 5}{x + 2}.\]

Key Concepts

Factoring Polynomials

In algebra, factoring polynomials is a vital skill, especially when dealing with rational expressions. The goal is to rewrite a polynomial as a product of its factors. This process simplifies solving equations and simplifies expressions.
To successfully factor a quadratic polynomial like the ones in our example,

• First, determine two numbers whose product is equal to the constant term (in the case of factoring quadratics, this is the product of the leading coefficient and the constant term) and whose sum is equal to the linear coefficient.
• Rewrite the middle term using these numbers, essentially decomposing it into two simpler terms.
• Group the terms into two pairs and factor each group separately.
• Finally, identify and factor out any common binomial factors, yielding the product of two binomials.

For example, in the numerator of our original expression, you start with the quadratic \(6x^2 - 7x - 5\). By identifying \(-10\) and \(3\) as the factors of \(-30\) that add to \(-7\), we can skillfully rewrite and factor the expression to \((2x + 1)(3x - 5)\).
This meticulous breakdown of the polynomial into simpler parts is what allows us to move forward with simplifying the rational expression.

Rational Expressions

Rational expressions are fractions containing polynomials in both the numerator and denominator. Simplifying these expressions requires similar steps to simplifying numeric fractions, such as finding the greatest common factor and canceling it out. A rational expression is considered simplified when no factors are shared between the numerator and the denominator.
Here is how you handle them:

• First, start by factoring both the numerator and the denominator into their simplest polynomial forms.
• Analyze the factors in both the top and the bottom to discover common factors.
• Cancel out the common factors to simplify the rational expression.

In our example, the original expression \(\frac{6x^2 - 7x - 5}{2x^2 + 5x + 2}\) gets simplified after factoring. Each part is decomposed to reveal common factors, \((2x + 1)\), which are then canceled out, leaving \(\frac{3x - 5}{x + 2}\).
This illustrates the harmonious process of reducing a complex expression to a more manageable form.

Algebraic Simplification

Algebraic simplification is the art of making complex mathematical expressions easier to work with. This is particularly useful when you need to solve algebraic equations, analyze expressions, or perform operations such as addition or multiplication. It involves a variety of tactics:

• Factoring expressions to remove common terms.
• Simplifying each component of the expression individually.
• Utilization of properties of arithmetic to combine and reduce terms.

In the context of our exercise, simplification involved both factoring the polynomials and identifying common terms that could be canceled out, ultimately resulting in an expression that was simpler in its form, \(\frac{3x - 5}{x + 2}\).
By following these simplification techniques, one can effectively work with and understand complicated algebraic expressions, making problem-solving a breeze.