Problem 32 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{x^{2}+7 x-8}{1+x-2 x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{(x-1)(x+8)}{-(2x-1)(x+1)}\).

Step 1: Factoring the numerator

Factoring is the process of breaking down a polynomial into its simplest form. The numerator of the given expression is a quadratic polynomial. Using the form \(ax^{2} + bx + c\), where a=1, b=7, and c=-8, we need to find two numbers that multiply to -8 (product of a and c) and add to 7 (b). Those two numbers are -1 and 8. So, the factored form of the numerator is \(x^{2}+7 x-8 = (x-1)(x+8)\).

Step 2: Factoring the denominator

The denominator of the given expression is also a quadratic polynomial. Using the form \(ax^{2} + bx + c\), where a=-2, b=1, and c=1, we need to find two numbers that multiply to -2 (product of a and c) and add to 1 (b). Those two numbers are -1 and 2. So, the factored form of the denominator is \(1+x-2 x^{2} = -(2x-1)(x+1)\). Notice the negative sign before the factored expression, this is because 'a' in the quadratic expression is negative.

Step 3: Simplifying the expression

Now we substitute the factored forms of the numerator and the denominator into the rational expression and cancel out the common factors. The given expression simplifies to \(\frac{x^{2}+7 x-8}{1+x-2 x^{2}} = \frac{(x-1)(x+8)}{-(2x-1)(x+1)}\). There are no common factors in this fraction, hence, this is the simplified form.

Key Concepts

Factoring

Factoring is an essential technique in algebra for simplifying complex polynomials. It involves breaking down a polynomial into a product of its simpler components, known as factors. This process is akin to finding the pieces that, when multiplied together, yield the original expression.

• When factoring a polynomial, you typically look for common factors or use methods like grouping.
• Factoring is crucial in solving equations since it often reveals roots of the polynomial.

An example, as seen in the exercise, is factoring the quadratic polynomial in the numerator. By identifying two numbers that both multiply to give the constant term and add up to the linear coefficient, you can express it in factored form. Factoring provides not just a way to simplify expressions, but it also helps reveal characteristics of the polynomial, such as its roots and graph's intersections.

Quadratic Polynomial

A quadratic polynomial is a polynomial expression where the highest power of the variable is two. It is generally expressed in the form: \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

• The quadratic polynomial's shape on a graph is a parabola, which opens upwards if \(a\) is positive and downwards if \(a\) is negative.
• The solutions or "roots" of the quadratic come from setting the polynomial equal to zero and solving—often through factoring, the quadratic formula, or completing the square.

In the given problem, both the numerator and the denominator are quadratic polynomials. They can be factored to simplify the rational expression, transforming the polynomials from standard to factored form, making it easier to see potential common terms that can be cancelled in later steps.

Simplifying Rational Expressions

Simplifying a rational expression involves reducing it to its simplest form by eliminating common factors from the numerator and denominator. This process is similar to simplifying a fraction.

• First, ensure that each part of the expression is factored correctly.
• Check for common factors between the numerator and denominator.
• Cancel out any common factors to simplify the expression to its lowest terms.

In our exercise, the expression was simplified by factoring both the numerator and the denominator of the fraction. After factoring, the simplified form was achieved by ensuring there were no common terms to cancel, thus ensuring the expression was in its simplest possible form.

Numerator and Denominator

In a fraction or rational expression, the numerator and denominator refer, respectively, to the top and bottom parts of the fraction.

• The numerator represents how many parts of a whole are considered or counted.
• The denominator indicates into how many parts the whole is divided.

When dealing with rational expressions, like in the exercise, the aim is to manipulate the numerator and denominator to provide the simplest form of the fraction. Factoring plays a key role in this process, as seen, by converting both parts into products of simpler binomials. This aids in identifying and eliminating any common factors, which is the key step in simplification. It's crucial to fully understand each part's role in order to manage complex expressions effectively.