Problem 65

Question

Find the product. $(2 x+7)(3 x-1)$

Step-by-Step Solution

Verified

Answer

The product of $(2 x+7)(3 x-1)$ is $6x^2 + 19x - 7$.

1Step 1: Apply FOIL Method - First

Multiply the first terms in each binomial: $2x * 3x$, which gives $6x^2$.

2Step 2: Apply FOIL Method - Outer

Multiply the outer terms: $2x * -1$, which gives $-2x$.

3Step 3: Apply FOIL Method - Inner

Multiply the inner terms: $7 * 3x$, which gives $21x$.

4Step 4: Apply FOIL Method - Last

Multiply the last terms: $7 * -1$, which gives $-7$.

5Step 5: Simplify the Expression

Combine all the terms from steps 1-4: $6x^2 - 2x + 21x - 7$. Now, combine like terms, the $-2x$ and $21x$ to give $6x^2 +19x -7$.

Key Concepts

FOIL MethodBinomialsDistributive Property

FOIL Method

The FOIL Method is a valuable shortcut for students multiplying two binomials. It stands for First, Outer, Inner, and Last. This outlines the sequence used to multiply the terms.

First: Multiply the first terms in each of the binomials. For example, with $(2x+7)$ and $(3x-1)$, multiply $2x$ and $3x$ to get $6x^2$.
Outer: Multiply the outermost terms. Here, $2x$ and $-1$ are outer terms, giving $-2x$.
Inner: Multiply the inner terms. With $7$ and $3x$, this results in $21x$.
Last: Finally, multiply the last terms. Multiply $7$ and $-1$ to get $-7$.

Once these four steps are done, add all the results together and combine any like terms to simplify. In this example, after adding together $6x^2$, $-2x$, $21x$, and $-7$, we combine $-2x$ and $21x$ to yield $6x^2 + 19x - 7$. This process makes it easier to correctly multiply and simplify binomials.

Binomials

A binomial is simply a type of polynomial with two terms. Each term is separated by a positive or negative sign. For example, in the problem $(2x + 7)$ and $(3x - 1)$, both are binomials. Each binomial contains two distinct terms.

The terms in a binomial can include:

Variables with coefficients, such as $2x$ or $3x$ in our example
Constants, such as the $+7$ and $-1$

When multiplying binomials, understanding each term plays a crucial role. It allows you to apply methods like FOIL properly. Also, recognizing that binomials are part of polynomials helps when dealing with more complex expressions. Whether for expansion or simplification, binomials lay a foundational math concept that students will encounter throughout algebra.

Distributive Property

The Distributive Property facilitates the process of expanding expressions. This property states that for any three numbers or algebraic expressions $a$, $b$, and $c$, the expression $a(b + c)$ translates into $ab + ac$. This principle is pivotal for multiplying binomials efficiently.

For the binomials $(2x+7)(3x-1)$, using the Distributive Property means distributing each term in the first binomial across each term in the second binomial.

First, $2x$ distributes over $3x$ and then over $-1$, producing $6x^2$ and $-2x$ respectively.
Next, the constant $+7$ distributes over both $3x$ and $-1$, resulting in $21x$ and $-7$.

Finally, by applying the Distributive Property systematically, you can combine results and simplify to get $6x^2 + 19x - 7$. This property is a cornerstone of algebra, offering a reliable approach to handle more complex multiplication tasks with ease.

Problem 65

Problem 66

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