Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\((2 x-1)(x+3)+3(x+3)\)

Step-by-Step Solution

Answer

The product of \((2 x-1)(x+3)+3(x+3)\) simplifies to \(2x^2 + 8x + 6\).

Step 1: Distribute the Binomial

Firstly, distribute the first binomial \((2x - 1)\) across the second binomial \((x + 3)\) which gets: \(2x^2 + 6x - x - 3\) or \(2x^2 + 5x - 3\)

Step 2: Distribute the Monomial

Next, distribute the monomial \(3\) across the binomial \((x + 3)\) which results in: \(3x + 9\)

Step 3: Combine Like Terms

Lastly, combine the expressions obtained in the first two steps: \(2x^2 + 5x - 3 + 3x + 9\). This simplifies to: \(2x^2 + 8x + 6\).

Key Concepts

Distributive Property

The distributive property is a bedrock mathematical principle, crucial when multiplying binomials or expanding polynomials. It allows for multiplication across each term within parentheses. For instance, when you have an expression like \(2x - 1)\times(x + 3)\), you apply the distributive property, sometimes known as the 'FOIL' method when dealing with binomials. This method involves multiplying each term of the first binomial, \(2x - 1)\), with every term of the second binomial, \(x + 3)\).

Through this process:

• The term \(2x\) multiplies with \(x\) to give \(2x^2\),
• then \(2x\) multiplies with \(3\) to provide \(6x\),
• the term \( -1\) multiplies with \(x\) resulting in \( -x\),
• and lastly, \( -1\) multiplies with \(3\) giving \( -3\).

Adding these together provides the expanded form \(2x^2 + 6x - x - 3\), which simplifies to \(2x^2 + 5x - 3\). This illustrates the distributive property's application: each term in one binomial must be distributed across all terms in the other binomial.

Combining Like Terms

Combining like terms is a way to streamline expressions, making them easier to handle by focusing on simplification. It involves the consolidation of terms within an expression that have the same variable to various powers. When you look at the expression \(2x^2 + 5x - 3 + 3x + 9\), combining like terms means looking for terms that have the same variables raised to the same power.

Here's what happens in our example:

• The \(5x\) and \(3x\) are like terms because they're both coefficients of \(x\) to the power of 1, and their combination gives \(8x\).
• The numbers \( -3\) and \(9\) are constants without variables, so these combine to make \(6\).

Consequently, the simplified form of the polynomial is \(2x^2 + 8x + 6\). This technique is central to manipulating and understanding algebraic expressions, enabling further operations such as addition, subtraction, and even factorization.

Polynomial Operations

Handling polynomial operations encompasses a scope of strategies, with binomial multiplication being a crucial aspect. The operation discussed here exemplifies the routine steps to take when manipulating polynomials: distributing and combining like terms. However, it's important to note that polynomial operations are not limited to these and can include addition, subtraction, multiplication, division, and factoring among others.

As you work through exercises like \(2x - 1)\times(x + 3) + 3(x + 3)\), you practice these operations extensively. Each step, from expanding the expression by distribution to simplifying by combining like terms, builds your skills in handling more complex polynomial problems. The key is to take systematic steps, ensuring each operation adheres to the algebraic principles, and thus arriving at the correct simplified form of polynomials.