Problem 43 - Elementary and Intermediate Algebra

Question

Simplify each rational expression. See Example 4 $$\frac{p^{3}+p^{2} q-2 p q^{2}}{p q^{2}+p^{2} q-2 p^{3}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(-1\).

Step 1: Identify the Numerator

The numerator of the expression is \( p^3 + p^2 q - 2pq^2 \). We need to factor this to simplify the expression.

Step 2: Factor the Numerator

Look for common factors in \( p^3 + p^2 q - 2pq^2 \). The greatest common factor (GCF) is \( p \). Factoring out \( p \), we get: \( p(p^2 + pq - 2q^2) \).

Step 3: Factor the Quadratic

Focus on \( p^2 + pq - 2q^2 \). To factor this quadratic, look for two numbers whose product is \(-2q^2\) and sum is \(q\). These numbers are \(2q\) and \(-q\), giving us: \( (p + 2q)(p - q) \). Therefore, the fully factored form of the numerator is \( p(p + 2q)(p - q) \).

Step 4: Identify and Factor the Denominator

The denominator is \( pq^2 + p^2 q - 2p^3 \). Factor out the greatest common factor of \( p \) from the terms: \( p(q^2 + pq - 2p^2) \).

Step 5: Factor the Quadratic in the Denominator

Address the quadratic \( q^2 + pq - 2p^2 \). We look for numbers that multiply to \(-2p^2\) and add to \(p\). The numbers \(2p\) and \(-p\) meet this requirement, so factor as: \( (q - p)(q + 2p) \). The fully factored denominator is \( p(q - p)(q + 2p) \).

Step 6: Simplify the Expression

We now have: \[\frac{p(p + 2q)(p - q)}{p(q - p)(q + 2p)}\]Cancel out common factors! The common factor is \( p \), reducing the expression to:\[\frac{(p + 2q)(p - q)}{(q - p)(q + 2p)}\]Notice that \( (q-p) = -(p-q) \), so we can simplify further to:\[-\frac{(p + 2q)}{(q + 2p)}\]

Step 7: Final Simplification

Finally, the expression reduces to:\(-1\).

Key Concepts

Factoring Polynomials

Factoring polynomials can feel like uncovering a hidden pattern in a puzzle. The process involves rewriting a polynomial as a product of its smaller polynomials or factors. This is akin to breaking down a complex system into simpler and more manageable parts. To factor a polynomial successfully, it's essential to identify one of its primary aspects - common factors and recognizing recognizable patterns such as a difference of squares or perfect square trinomials.
A good starting point in factoring is always to check if there is a greatest common factor (GCF) that you can factor out first. Once this is done, tackling the quadratic expressions becomes much easier. So, factoring not only helps simplify expressions but also uncovers deeper insights into the nature of the polynomial itself.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is a key concept when simplifying rational expressions, particularly in expressions with multiple terms. It refers to the highest factor that divides each term of the polynomial without leaving a remainder. Determining the GCF is often the first step in the factoring process.
For the polynomial in question, you identify a common factor present in each of the terms. Once the GCF is identified, it can be factored out of the polynomial, simplifying and reducing the complexity of the expression.

• Finding the GCF can bring down large and unwieldy polynomials into smaller, more workable forms.
• Factoring the GCF is crucial also because it sets the stage for further factoring of the reduced polynomial, leading to complete simplification.

It's important always to remind yourself that factoring out the GCF should be done before attempting to tackle more specific factoring techniques.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically in the form of \( ax^2 + bx + c \). Understanding them is crucial as they are a common type of expression you'll encounter in algebra. When it comes to simplifying rational expressions, recognizing and factoring quadratic expressions is vital.
To factor a quadratic, you look for two numbers whose product equals the constant term (\( ac \)), and whose sum equals the linear coefficient (\( b \)). This technique often involves trial and error or utilizing patterns of common factorable forms.

• For instance, a familiar pattern might be that of a trinomial, which can be factored into binomials like \( (x + m)(x + n) \).
• Recognizing and exploiting symmetries and patterns in these expressions can simplify the process significantly.

By understanding quadratic expressions, you enable yourself to not only simplify complex rational expressions but also solve quadratic equations efficiently.