Problem 88 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{6 x^{2}+13 x+6}{6-5 x-6 x^{2}} $$

Step-by-Step Solution

Answer

Simplifies to -1.

Step 1: Identify Forms

Recognize that both the numerator and the denominator of the expression are quadratic polynomials. Specifically, the numerator addresses the standard form \( ax^2 + bx + c \) and so does the denominator albeit in reverse order.

Step 2: Rearrange Denominator

Reorder the terms of the denominator from \( 6 - 5x - 6x^2 \) to \( -6x^2 - 5x + 6 \) for consistent comparison with the numerator.

Step 3: Factor the Numerator

Factor \( 6x^2 + 13x + 6 \) by seeking two numbers that multiply to \( 6 \times 6 = 36 \) and add to \( 13 \). These numbers are \( 4 \) and \( 9 \). Thus, we can write the numerator as \((2x + 3)(3x + 2)\).

Step 4: Factor the Denominator

Factor \(-6x^2 - 5x + 6\) similarly. Here, find two numbers that multiply to \(-6 \times 6 = -36\) and add to \(-5\). These numbers are \(-9\) and \(4\). Thus, the denominator can be expressed as \(-(2x + 3)(3x + 2)\).

Step 5: Cancel Common Factors

Notice both the numerator and denominator have common binomial factors \((2x + 3)\) and \((3x + 2)\). Cancel these to simplify the expression to \(-1\) after accounting for the negative sign in the denominator's expression.

Key Concepts

Factoring Quadratic Polynomials

Factoring quadratic polynomials is a key skill in algebra that involves breaking down a quadratic expression into the product of two binomials. In general, a quadratic polynomial takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The goal is to express this quadratic as \((mx + n)(px + q)\), where \( m \), \( n \), \( p \), and \( q \) are constants that must be determined.

To do this, look for two numbers that multiply to \( a \times c \) and add to the coefficient \( b \) of the middle term. For example, for \( 6x^2 + 13x + 6 \), you need numbers that multiply to 36 and add to 13. These numbers are 4 and 9. This allows us to rewrite the expression by splitting the middle term:

• Replace 13x with 4x + 9x.
• Group and factor by grouping.

Finally, recognize common factors in these grouped terms to write the expression as \((2x + 3)(3x + 2)\). Through this process, quadratic polynomials become much easier to handle, especially when found in fractions that need simplifying.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This is often key in making math problems easier to solve or understand. A more simplified expression is easier to work with in subsequent steps of problem-solving.

For rational expressions like \(\frac{6x^2 + 13x + 6}{6 - 5x - 6x^2}\), first, factor both the numerator and the denominator when possible.

• This allows us to find and "cancel" identical terms that appear in both, leaving us with a simpler expression.
• Remember, you can only cancel factors (entire multiplied parts), not terms within an addition or subtraction.

In our example, both numerator and denominator factored to the same binomials, allowing us to simplify down to \(-1\) because of an initial negative sign. Always be cautious about this negative factor in the denominator, as it affects the overall sign after canceling common factors.

Polynomial Arithmetic

Polynomial arithmetic includes operations such as addition, subtraction, multiplication, and division of polynomials. When dealing with division, particularly within rational expressions, proficiency in polynomial arithmetic becomes crucial.

To handle problems like \(\frac{6x^2 + 13x + 6}{6 - 5x - 6x^2}\), the objective is to factor the quadratic polynomials in both the numerator and the denominator.

• This transformation into a product of binomials facilitates the division process.
• After factoring, you can manage the arithmetic part by eliminating common factors, providing a tidy and simplified result.

Even though both parts of the expression might look complex initially, recognizing patterns and applying basic operations can substantially reduce your work. It is a fundamental tactic in algebra to condense expressions wherever possible to simplify your entire calculation to its most basic essence.