Find each product. See Example 5. (3x + 1) (2x - 7)

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Identify the expression to be multiplied: $(3x + 1)(2x - 7)$.

Apply the distributive property (also known as FOIL for binomials) to multiply each term in the first binomial by each term in the second binomial: multiply \$3x$ by $2x$, then $3x$ by $-7$, then \(1$ by \)2x$, and finally \(1$ by \)-7$.

Write out each product explicitly: $3x \times 2x = 6x^2$, $3x \times (-7) = -21x$, $1 \times 2x = 2x$, and $1 \times (-7) = -7$.

Combine all these products into a single expression: \$6x^2 - 21x + 2x - 7$.

Simplify the expression by combining like terms: combine $-21x$ and \$2x$ to get the final simplified polynomial.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distributive Property

The distributive property allows you to multiply a single term by each term inside a parenthesis. For example, in (a + b)(c + d), you multiply each term in the first parenthesis by each term in the second. This property is essential for expanding products of binomials.

Multiplying Binomials

Multiplying binomials involves applying the distributive property twice or using the FOIL method (First, Outer, Inner, Last) to multiply each pair of terms. This process results in a polynomial expression that combines like terms.

Combining Like Terms

After multiplying terms, you often get several terms with the same variable and exponent. Combining like terms means adding or subtracting these terms to simplify the expression into its simplest polynomial form.

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