Problem 59 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 2 b(b-1) $$

Step-by-Step Solution

Answer

Question: Expand and simplify the following expression: \(2b(b-1)\) Answer: \(2b^2 - 2b\)

Step 1: Apply the distributive property

To expand the given expression \(2b(b-1)\), we need to apply the distributive property which states that \(a(b+c) = ab + ac\). In this case, we will multiply \(2b\) with each term inside the parentheses. $$ 2b(b-1) = 2b \cdot b + 2b \cdot (-1) $$

Step 2: Perform the multiplication

Now we will perform the multiplication of the constants and variables. $$ 2b \cdot b + 2b \cdot (-1) = 2b^2 - 2b $$

Step 3: Check for any like terms

In the expanded expression \(2b^2 - 2b\), there are no like terms to combine. A like term would be terms having the same variable and exponent. So, the expression is already simplified: $$ 2b(b-1) = 2b^2 - 2b $$

Key Concepts

Multiplying Polynomials

The process of multiplying polynomials involves the distributive property of algebra, where you multiply each term in one polynomial by each term in the other polynomial. This is often known by its more common name: the FOIL method, standing for First, Outer, Inner, Last, which applies to the multiplication of binomials.

For our example, we have the expression \(2b(b-1)\). To multiply this polynomial, we distribute \(2b\) across the terms within the parentheses:

• First, we multiply \(2b\) by \(b\), yielding \(2b^2\).
• Next, we multiply \(2b\) by \( -1\), which gives us \( -2b\).

The result is simply the sum of these individual products, giving us \(2b^2 - 2b\).

Combining Like Terms

Combining like terms is a fundamental part of simplifying algebraic expressions. Like terms are terms in an expression that have the same variables raised to the same exponents. They can be combined by adding or subtracting their coefficients.

In the expression we obtained from multiplying the polynomials, \(2b^2 - 2b\), there are two terms but they are not like terms because they do not have the same powers of\b). \(2b^2\) is a term with \(b\) squared, and \( -2b\) has \(b\) to the first power. Since there are no like terms, this expression is already in its simplest form and no further combination is required.

Algebraic Expression Simplification

Simplification of an algebraic expression involves reducing it to its most basic form without changing its value. This often includes expanding expressions using the distributive property, combining like terms, and factoring, where applicable.

The goal is to make the expression as straightforward as possible, both for ease of understanding and solving equations. For the expression \(2b(b-1)\), after expanding and combining like terms, we are left with \(2b^2 - 2b\). There are no further simplifications or factoring possibilities, and thus we conclude that this is the simplest form of the given algebraic expression.