Problem 79 - Algebra 1

Question

Find the product. $$(x-2)(x+11)$$

Step-by-Step Solution

Answer

The product of \( (x-2) \) and \( (x+11) \) is \( x^2 + 9x - 22 \)

Step 1: Apply Distributive Law - Part 1

Multiply the first term in the first binomial by each term in the second binomial.\n So, this will give us: \( x * (x + 11) = x^2 + 11x \)

Step 2: Apply Distributive Law - Part 2

Multiply the second term in the first binomial by each term in the second binomial.\n This will give us: \( -2 * (x + 11) = -2x - 22 \)

Step 3: Combine Like Terms

Add what was calculated in Step 1 and Step 2 together.\n Therefore, we add \( x^2 + 11x \) and \( -2x - 22 \) together to get the final product, which is \( x^2 + 9x - 22 \)

Key Concepts

Distributive Property

The Distributive Property is a key tool in algebra, especially when we're working with binomials like \((x-2)(x+11)\). This property allows us to simplify expressions by distributing or spreading out the terms. When we say 'distribute,' we mean that each term inside one binomial is multiplied by each term in the other binomial.Think of the Distributive Property as unpacking each binomial into individual terms, then multiplying them step by step. In our case, we have two binomials:

• \(x\) in \((x-2)\) is multiplied by every term \((x+11)\).
• \(-2\) in \((x-2)\) is also multiplied by every term \((x+11)\).

This creates the smaller expressions \(x^2 + 11x\) and \(-2x - 22\). Without the Distributive Property, combining the binomials would be nearly impossible.

Combining Like Terms

Once we've distributed every term, we're often left with a combination of terms that can be simplified by combining like terms. Combining like terms means adding or subtracting coefficients of terms with the same variable and power.For instance, after distributing, we arrived at:

• \(x^2 + 11x\)
• \(-2x - 22\)

The 'like' terms here are \(11x - 2x\) because they both have the variable \(x\) raised to the same power, which is 1. This allows us to merge them into one single term: - Combining \(11x - 2x\) results in \(9x\).Combining these like terms simplifies the entire expression to \(x^2 + 9x - 22\). Remember, combining like terms reduces complexity and makes the expression easier to understand and work with.

Polynomial Expressions

Polynomial expressions, such as \((x-2)(x+11)\), are composed of terms that can include constant numbers, variables, and the exponents of these variables. The expression \(x^2 + 9x - 22\) is a product of combining terms derived from binomial multiplication.A polynomial is defined by:

• **Terms**: Parts of the expression separated by plus or minus signs. Here, \(x^2, 9x,\) and \(-22\) are the terms.
• **Degree**: Related to the highest power of the variable; for this polynomial, the highest power is 2 (from \(x^2\)), so it's a quadratic polynomial.
• **Variables and Coefficients**: Variables are the letters (e.g., \(x\)), and coefficients are numbers that multiply the variables (e.g., 1 in \(x^2\), 9 in \(9x\)).

Understanding the structure of polynomial expressions helps with the manipulation and solving of algebraic equations. Recognizing polynomial patterns is essential in algebra to identify how these can be simplified into a clear and workable form.