Problem 19 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{3 a}{a+4} \cdot \frac{a^{2}+5 a+4}{a^{2}+a}$$

Step-by-Step Solution

Answer

The simplest form of the given expression is \(3\)

Step 1: Factorize the Expressions

First, rewrite each of the numerator and denominator polynomials as their factorized forms. The expressions \(a^{2}+5 a+4\) and \(a^{2}+a\) can be factorized. \(a^{2}+5 a+4\) can be written as \((a+1)(a+4)\) and \(a^{2}+a\) can be written as \(a(a+1)\). Now replace these factorized expressions in the original problem. \(\frac{3 a}{a+4} \cdot \frac{(a+1)(a+4)}{a(a+1)}\)

Step 2: Simplify the Expression

In the fraction multiplication, simplify by canceling out the like terms from the numerator and denominator. Notice the \(a+4\) term and the \(a+1\) term both appear in the numerator and denominator, hence they can be canceled out. So, the simplified expression becomes \(3 \cdot \frac{a}{a}\)

Step 3: Final Simplification

Simplify it further by canceling out \(\frac{a}{a}\) which is equal to 1. So, the final simplified product is \(3\)

Key Concepts

Understanding Polynomial Factorization

Factoring polynomials is a technique used to simplify expressions by writing them as a product of their simpler building blocks, called factors. This is similar to breaking down numbers into prime factors in arithmetic. For example, the expression \(a^2 + 5a + 4\) is a polynomial that can be factorized into \((a+1)(a+4)\).

• The first step is to identify two numbers that multiply to the constant term (here it's 4) and add up to the linear coefficient (here it's 5).
• These two numbers are 1 and 4.
• Thus, the polynomial can be rewritten as \((a+1)(a+4)\).

Similarly, the polynomial \(a^2 + a\) can be factorized as \(a(a+1)\). By identifying common factors in complicated expressions, we simplify them greatly. This makes subsequent operations like multiplication or division much easier.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves cancelling out common factors in the numerator and denominator. In the context of algebraic fractions, this also includes polynomial terms.Take a look at the fraction \(\frac{3a}{a+4} \cdot \frac{(a+1)(a+4)}{a(a+1)}\). After factorization, you might find lterms both in the numerator and denominator.

• Here, \(a+4\) and \(a+1\) appear on both sides.
• Cancelling them results in \(3 \cdot \frac{a}{a}\).

After canceling \(a/a\), you're left with a much more manageable expression. Through simplification, complex algebraic expressions become significantly easier to handle.

Multiplication of Fractions

Multiplying fractions involves finding the product of the numerators and the product of the denominators separately. In algebra, this principle applies to algebraic fractions as well.For example, with two fractions: \(\frac{a}{b}\) and \(\frac{c}{d}\), multiply the numerators \(a\) and \(c\) together to get the new numerator \(a \cdot c\), and the denominators \(b\) and \(d\) to get the new denominator \(b \cdot d\). Applying this to the initial problem before simplification:

• Numerator: \(3a \cdot (a+1)(a+4)\)
• Denominator: \((a+4) \cdot a(a+1)\)

After the simplification steps, the result is simply \(3\). The process highlights that multiplication of fractions requires attentive simplification to ensure the simplest form of the expression is reached. This makes it easier to work with and understand.