Question

Multiply the polynomials using the FOIL method. Express your answer as a single polynomial in standard form. $$ (3 x+1)(2 x+1) $$

Short answer

6x^2 + 5x + 1

Step-by-step solution

Apply the FOIL Method

The FOIL method stands for First, Outer, Inner, Last. Multiply the first terms, outer terms, inner terms, and last terms of the binomials.

Multiply the First Terms

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

Multiply the Outer Terms

Multiply the outer terms of each binomial: (3x) * (1) = 3x

Multiply the Inner Terms

Multiply the inner terms of each binomial: (1) * (2x) = 2x

Multiply the Last Terms

Multiply the last terms of each binomial: (1) * (1) = 1

Combine Like Terms

Add all the results from the previous steps: 6x^2 + 3x + 2x + 1. Combine like terms: 6x^2 + 5x + 1

Key concepts

Polynomial Multiplication

Multiplying polynomials involves distributing each term in the first polynomial to every term in the second polynomial. There are several methods to multiply polynomials, but one popular technique for binomials is the FOIL method. FOIL stands for First, Outer, Inner, Last. This method organizes the multiplication process and helps ensure all terms are accounted for.

For instance, consider the exercise \( (3x+1)(2x+1) \). Using FOIL, we multiply terms in a specific order:

• First: Multiply the first terms of each binomial \( 3x \times 2x = 6x^2 \)
• Outer: Multiply the outer terms \( 3x \times 1 = 3x \)
• Inner: Multiply the inner terms \( 1 \times 2x = 2x \)
• Last: Multiply the last terms \( 1 \times 1 = 1 \)

After performing these steps, we combine the results to get the final polynomial.

Binomials

A binomial is a polynomial that consists of exactly two terms. In the given exercise, both \( 3x+1 \) and \( 2x+1 \) are binomials. When multiplying binomials, each term in the first binomial must be multiplied by each term in the second binomial, which is effectively what the FOIL method does.

Understanding binomials is crucial as they are often simpler to work with and appear frequently in algebraic expressions. They serve as building blocks in more complex polynomial operations and factor attempts.

• Trick: Whenever you see an expression with two terms added or subtracted, you are dealing with a binomial.
• Examples: Binomials can have constants or variables, such as \(x+5\) or \(2y-3\).
• Note: The presence of exponents doesn't affect the binomial identity; \(x^2 + 3x \) is still a binomial.

Combining Like Terms

Combining like terms is a necessary step after performing polynomial multiplication. Like terms are terms that have the same variable raised to the same power. In the expression obtained from our FOIL method, \( 6x^2 + 3x + 2x + 1 \,\), we identify like terms and add them:

• \