Problem 15 - Elementary Algebra

Question

Multiply. (Assume all denominators are nonzero.) $$ x+1 x-3 \cdot 3-x x+5 $$

Step-by-Step Solution

Answer

The simplified result is \(-x^2 - 6x - 5\).

Step 1: Identify and Rewrite the Expressions

Begin by identifying each binomial and rewriting the expression using parentheses to represent multiplication: \[(x+1)(x-3) \cdot (3-x)(x+5)\]Recognize that \(3-x\) is the same as \(-(x-3)\).

Step 2: Simplify Using Identity

Recognize \(3-x = -(x-3)\), which allows us to rewrite \[(x+1)(x-3) \cdot -(x-3)(x+5)\]This provides an opportunity to cancel out the \((x-3)\) terms.

Step 3: Cancel Out Common Terms

Cancel the common term \((x-3)\) in both the numerator and denominator: \[(x+1) \cdot -(x+5)\]Now, you are left with two binomials to multiply.

Step 4: Multiply the Remaining Binomials

Distribute to multiply the remaining binomials: \[(x+1) \cdot -(x+5) = -[(x \cdot x + x \cdot 5 + 1 \cdot x + 1 \cdot 5)]\]Combine like terms from the expanded expression.

Step 5: Combine Like Terms

Combine the terms obtained from the multiplication: \[-(x^2 + 5x + x + 5) = -(x^2 + 6x + 5)\]Distribute the negative sign across the polynomial.

Step 6: Simplify Final Expression

Apply the negative sign to the polynomial: \[-x^2 - 6x - 5\]This is the final simplified result of the multiplication of the original expression.

Key Concepts

binomial multiplication

Binomial multiplication is a foundational concept in algebra that involves multiplying two expressions, each containing two terms, also known as binomials. This operation is commonly seen in expressions like \((x + 1)(x - 3)\). It requires the distributive property, often remembered by the acronym FOIL, which stands for First, Outside, Inside, Last. Here's how it works:

• First: Multiply the first terms of each binomial.
• Outside: Multiply the outer terms in the multiplication.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

When multiplying binomials, you perform these steps, then combine like terms to simplify the expression. For example, multiplying \((x + 1)(-x + 5)\) involves calculating each product:- First: \(x \cdot -x = -x^2\)- Outside: \(x \cdot 5 = 5x\)- Inside: \(1 \cdot -x = -x\)- Last: \(1 \cdot 5 = 5\)
By rearranging and combining like terms, you obtain the expression \(-x^2 + 4x + 5\). This method helps in systematically finding the product of binomials and preparing them for further steps like factoring or simplification.

factoring

Factoring is the process of finding two or more expressions that, when multiplied together, produce a given expression. It is a useful skill for simplifying polynomials and solving equations. In the provided exercise, recognizing that terms can factor into simpler binomials, like \((3-x)\) reorganizing as \(-(x-3)\), is key in the solution. This presents an opportunity to cancel terms, making the expression easier to work with.

• Identifying Common Factors: Look for terms that can be factored out. For instance, recognizing that \(3-x\) can be rewritten as \(-1(x-3)\).
• Canceling Out: Once common factors are identified, they can often be canceled if they appear in both the numerator and denominator of a fraction. This was illustrated in the initial steps where \((x-3)\) was canceled.

Factoring can significantly simplify complex expressions, allowing for easier solutions and manipulation in algebraic equations. It's a strategic tool in algebra that assists students in solving and simplifying expressions efficiently.

polynomial simplification

Polynomial simplification involves reducing expressions to their simplest form by combining like terms and performing arithmetic operations. After multiplying binomials or factoring expressions, it is important to combine all like terms and simplify. In this exercise, after multiplying \((x+1)\) and \(-(x+5)\), the next step is to simplify:

• Combine Like Terms: After expansion, gather similar terms, such as terms containing \(x\), \(x^2\), or constants. In our example, \(-x^2, -6x,\) and \(-5\) are combined to yield \(-x^2 - 6x - 5\).
• Apply the Negative Sign: Carefully distribute negative signs across terms to ensure accuracy.

Achievement of polynomial simplification ensures that expressions are easier to work with and reflects the final, most reduced form of the mathematical sentence. It also demonstrates an expression's equivalence to other forms after applying algebraic operations.