Problem 56 - College Algebra with Modeling and Visualization

Question

Multiply the binomials. $$(3 x+1)(2 x+1)$$

Step-by-Step Solution

Answer

The product is \(6x^2 + 5x + 1\).

Step 1: Identify the Binomials

The given expression is the product of two binomials: \((3x + 1)\) and \((2x + 1)\). Our task is to multiply these two binomials together.

Step 2: Use the FOIL Method

To multiply two binomials, we use the FOIL method. FOIL stands for First, Outer, Inner, and Last, which refers to the pairs of terms you multiply together. Apply the method step by step.

Step 3: First Terms

Multiply the first terms of each binomial: \(3x \cdot 2x = 6x^2\).

Step 4: Outer Terms

Multiply the outer terms of the binomials: \(3x \cdot 1 = 3x\).

Step 5: Inner Terms

Multiply the inner terms of the binomials: \(1 \cdot 2x = 2x\).

Step 6: Last Terms

Multiply the last terms of each binomial: \(1 \cdot 1 = 1\).

Step 7: Combine All Products

Add all the products together from the FOIL method: \(6x^2 + 3x + 2x + 1\).

Step 8: Simplify the Expression

Combine like terms: \(6x^2 + (3x + 2x) + 1 = 6x^2 + 5x + 1\). This is the final simplified product of the binomials.

Key Concepts

FOIL Method

The FOIL method is a handy tool that simplifies the process of multiplying two binomials. This technique is a mnemonic that stands for First, Outer, Inner, and Last. It guides us on which terms to multiply together when facing binomial multiplication problems.

• First: Multiply the first terms in each binomial. For our example, \(3x\) and \(2x\) are the first terms, yielding \(6x^2\).
• Outer: Next, multiply the outer terms of the binomials which are \(3x\) and \(1\). The result is \(3x\).
• Inner: Then, multiply the inner terms; \(1\) and \(2x\) in our problem, producing \(2x\).
• Last: Finally, multiply the last terms of each binomial. Here, that’s \(1\) and \(1\), resulting in \(1\).

After applying all FOIL steps, ensure to combine like terms to get the final expression. In this problem, combining gives us \(6x^2 + 5x + 1\). This method is invaluable for quickly tackling binomial products.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition or subtraction. Each term in a polynomial can have a variable raised to a whole number exponent. The highest power of the variable is known as the degree of the polynomial.

In binomial multiplication using the FOIL method, resultants are polynomial expressions. Our exercise results in the polynomial \(6x^2 + 5x + 1\). This polynomial is of degree 2 because the highest power of \(x\) is 2.

Polynomials play a key role in various areas of algebra and beyond, as they can model real-world situations and create equations to solve particular problems.

• Constant term: A term without any variable, like \(1\) in this polynomial.
• Linear term: A term with the variable raised to the first power, such as \(5x\).
• Quadratic term: A term with the variable raised to the second power, like \(6x^2\).

Understanding polynomials is fundamental for studying algebra and helps connect to more complex concepts.

Algebraic Expressions

Algebraic expressions use variables to represent unknown values, allowing us to form equations and solve problems. These expressions can include numbers and letters, which stand for numbers and operations.

When multiplying binomials to form a polynomial, the expression remains algebraic as we don't assign specific values to the variables involved. For example, \(3x + 1\) and \(2x + 1\) are algebraic expressions. Upon multiplication, we derive another algebraic expression: \(6x^2 + 5x + 1\).

• They can be simple, like \(x + 1\), or more complex, involving multiple terms and operations.
• Unlike numerical expressions, variables in algebraic expressions add flexibility for a range of values.

Algebraic expressions are essential for developing equations and are used widely in mathematics to solve for unknowns. They form the foundation for algebra and help in constructing models for real-life scenarios.