Problem 42 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 4. $$ (2 a x-10 x+a-5) \cdot \frac{x}{2 x^{2}+x} $$

Step-by-Step Solution

Answer

Multiply each term by \(x\), factor the denominator, then simplify resulting terms.

Step 1: Distribute the Numerator

The problem involves multiplying the polynomial \(2ax - 10x + a - 5\) by the fraction \(\frac{x}{2x^2 + x}\). We start by distributing the polynomial across the fraction's numerator, which is the same as multiplying each term in the polynomial by \(x\). We get:\[ x(2ax) - x(10x) + x(a) - x(5) = 2ax^2 - 10x^2 + ax - 5x \]

Step 2: Examine the Denominator

The denominator of the fraction \(2x^2 + x\) has a common factor. Factor out an \(x\):\[ 2x^2 + x = x(2x + 1) \]

Step 3: Simplify the Fraction

Now that we have the expression \(\frac{2ax^2 - 10x^2 + ax - 5x}{x(2x + 1)}\), divide each term in the numerator by \(x\): \[ \frac{2ax^2}{x} - \frac{10x^2}{x} + \frac{ax}{x} - \frac{5x}{x} = 2ax - 10x + a - 5 \] However, each distinct term in the simplified numerator cannot all be simplified to zero, refer back to the expression structure to verify no cancellation happens between the numerator and denominator after this simplification.

Key Concepts

Polynomial Multiplication

When we talk about polynomial multiplication, we are essentially discussing the process of distributing each term in one polynomial across each term in another expression. In this context, we started with a polynomial, \(2ax - 10x + a - 5\), and we needed to multiply it by a fraction, where the fraction's numerator is simply \(x\). This means for each term inside the polynomial, we multiply it by \(x\), or simply distribute \(x\) across the polynomial.

• This results in terms like \(x(2ax)\), \(x(-10x)\), \(x(a)\), and \(x(-5)\).
• Essentially, you multiply the coefficients and the variables separately. For example, \(x(2ax) = 2ax^2\) because \(x \cdot ax = ax^2\).
• This leads to the expanded polynomial \(2ax^2 - 10x^2 + ax - 5x\). Each of these represents the product from distributing \(x\) to the initial polynomial's terms.

Polynomial multiplication repeatedly comes in handy, especially in operations involving expressions and simplifying complicated algebraic forms.

Factoring Polynomials

Factoring is a key skill when working with polynomials, especially for simplifying expressions or solving equations. It involves expressing a polynomial as the product of its factors. In this problem's context, the denominator \(2x^2 + x\) was considered for factoring.

• We identified that a common factor between both terms of the denominator is \(x\).
• By factoring out \(x\), the expression \(2x^2 + x\) was transformed into \(x(2x + 1)\).
• Factoring helps simplify expressions by breaking them into simpler, multiplied forms, revealing commonalities that allow further simplification or cancellation.

Understanding factoring is vital, as it is not only a tool for simplification but is also essential for other processes, such as solving polynomial equations.

Fraction Simplification

Once you have multiplied polynomials and factored necessary components, the next logical step is to simplify the resulting fraction, if possible. Simplification involves reducing the fraction to its smallest possible form by cancelling common factors from the numerator and the denominator.

• In this case, our numerator \(2ax^2 - 10x^2 + ax - 5x\) was divided by every term in a common factor \(x\) from the denominator.
• Each term individually worked out to form \(2ax - 10x + a - 5\) upon normalization of \(x\).
• However, simplification didn’t lead to full cancellation as no further common factors were in both numerator and denominator.

The simplification of fractions is crucial as it makes the expression more manageable and easier to interpret or solve. Keep an eye out for opportunities to factor and reduce fractions whenever possible.