Problem 67 - Algebra 1: Concepts and Skills

Question

Find the product. \((3 x+1)(8 x-3)\)

Step-by-Step Solution

Answer

The product of \((3x+1)\) and \((8x-3)\) is \(24x^2 - x - 3\)

Step 1: Multiplication of First terms

Multiply the first terms in each binomial, which are \(3x\) and \(8x\). This gives \(3x * 8x = 24x^2\)

Step 2: Multiplication of Outside terms

Next, multiply the outside terms, which are \(3x\) and \(-3\). This results in \(3x * -3 = -9x\)

Step 3: Multiplication of Inside terms

Then, multiply the inside terms, which are \(1\) and \(8x\). This results in \(1 * 8x = 8x\)

Step 4: Multiplication of Last terms

Lastly, multiply the last terms, \(1\) and \(-3\). This gives \(1 * -3 = -3\)

Step 5: Sum of terms

Now, add together all four results from the previous steps. The sum of these products is the expanded form of the original expression: \(24x^2 - 9x + 8x -3 = 24x^2 - x - 3\)

Key Concepts

Binomial Multiplication

Binomial multiplication is a core principle of algebra and involves multiplying two expressions that each contain two terms. In the given problem, we have two binomials,

• The first binomial is \(3x + 1\)
• The second binomial is \(8x - 3\)

To find the product of these binomials, we use a systematic approach called the FOIL method, which stands for multiplying the First, Outside, Inside, and Last terms. This technique ensures you multiply every term in the first binomial by every term in the second binomial. Following the FOIL method guarantees that all possible combinations of terms are included in the product, which in this case results in four main partial products that need to be added together to form the final answer.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In our original exercise, the expressions \(3x + 1\) and \(8x - 3\) are both examples of binomials, which are specific types of algebraic expressions consisting of two terms. Each term in these binomials reflects an essential component:

• The term \(3x\) includes a coefficient (3) and a variable (x), indicating multiplication between them.
• The term \(1\) is a constant, meaning it remains the same regardless of the value of x.
• The second binomial works similarly with \(8x\) and \(-3\).

It's important when manipulating these expressions to respect the relationship and operation symbols, especially paying attention to signs when multiplying or adding terms.

Distributive Property

The distributive property is a fundamental rule in algebra that helps simplify expressions and solve equations. It states that multiplying a sum by a number is the same as doing each multiplication separately. This can be written as:\( a(b + c) = ab + ac \)In the context of our binomial multiplication exercise, this property ensures that every term in the first binomial multiplies by every term in the second one. When we distribute \(3x\) from the binomial \(3x + 1\) across the \(8x - 3\) binomial, we apply the distributive property like this:

• First, \(3x\) multiplies with \(8x\)
• Then, \(3x\) multiplies with \(-3\)

Similarly, \(1\) from the first binomial applies to each term in the second. This process results in four products that can be combined for the final expanded polynomial.