Problem 15 - Elementary Algebra

Question

For Problems \(11-36\), find the indicated products by applying the distributive property and combining similar terms. Use the following format to show your work: $$ \begin{aligned} (x+3)(x+8) &=x(x)+x(8)+3(x)+3(8) \\ &=x^{2}+8 x+3 x+24 \\ &=x^{2}+11 x+24 \end{aligned} $$ $$ (x-7)(x+1) $$

Step-by-Step Solution

Answer

The product is \(x^2 - 6x - 7\).

Step 1: Apply Distributive Property for the First Term

To find the product of \((x-7)(x+1)\), you first apply the distributive property. Distribute the first term \(x\) across the second binomial: \(x(x) + x(1)\). This gives \(x^2 + x\).

Step 2: Apply Distributive Property for the Second Term

Next, take the second term from the first binomial, which is \(-7\), and distribute it across the second binomial: \(-7(x) + (-7)(1)\). This results in \(-7x - 7\).

Step 3: Combine Like Terms

Now, combine all the terms you've calculated: \(x^2 + x - 7x - 7\). To simplify, identify and combine like terms: \(x^2 + (x - 7x) - 7\), leading to \(x^2 - 6x - 7\).

Key Concepts

Combining Like Terms

Combining like terms is a crucial step in simplifying expressions, especially in polynomial problems. In the expression \(x^2 + x - 7x - 7\), like terms are terms that have the same variable raised to the same power. Here, \(x\) and \(-7x\) are like terms because they both have the variable \(x\) raised to the first power.

• Identify the like terms: In our expression, this means recognizing \(x\) and \(-7x\) as terms that can be combined.
• Combine by adding or subtracting: Perform the operation \(x - 7x\), which results in \(-6x\).

This simplification leads to the final expression of \(x^2 - 6x - 7\). By combining like terms effectively, you reduce the complexity of the polynomial, making it easier to manage and solve.

Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial to every term of another. In the example \((x-7)(x+1)\), this process is essentially employing the distributive property multiple times.

• Distribute each term: Start by multiplying \(x\) by both \(x\) and \(1\), yielding \(x^2 + x\).
• Move to the next term: Then multiply \(-7\) by both \(x\) and \(1\), resulting in \(-7x - 7\).

After multiplication, you should have a set of terms that you can simplify by combining like terms. This method is not only applicable for simple binomials but also for more complex polynomials, making it an essential skill in algebra.

Binomials

A binomial is a polynomial with exactly two terms, such as \(x - 7\) or \(x + 1\). The goal is to understand the structure and behavior of these expressions, especially when multiplying them.

• Consist of two terms: Each term can be either a constant, a variable, or a product of both.
• Key in operations: Multiplying binomials is foundational to understanding more complex polynomial multiplication.
• Use of distributive property: The distributive property is often used to expand the product of binomials into a polynomial.

Knowing the basics of binomials allows for a more straightforward transition into more complex algebraic operations, given that they form the building blocks of polynomial arithmetic.