Problem 53 - Algebra A Combined Function

Question

Multiply vertically. \((2 x-11)(6 x+1)\)

Step-by-Step Solution

Answer

12x^2 - 64x - 11

Step 1: Apply the Distributive Property

To multiply two binomials, we start by applying the distributive property. This involves multiplying each term in the first binomial by each term in the second binomial. Write down these individual products. Multiply: 1. First terms: \( (2x) \times (6x) = 12x^2 \)2. Outer terms: \( (2x) \times (1) = 2x \)3. Inner terms: \((-11) \times (6x) = -66x \)4. Last terms: \((-11) \times (1) = -11 \)

Step 2: Combine Like Terms

Combine the like terms from step 1 to get the final expression. Since the only like terms involve \(x\), we combine them:\( 2x - 66x = -64x \).So, the expression is now: \[ 12x^2 - 64x - 11 \].

Step 3: Write the Final Expression

Write the simplified final expression after combining the like terms. The final product of the multiplication of the two binomials is:\[ 12x^2 - 64x - 11 \].

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps simplify expressions. It allows us to multiply a sum or difference by distributing the multiplier to each term inside the parentheses. This means taking each term of the first binomial and multiplying it with every term of the second binomial. In the context of our exercise,
we have:

• The first binomial: \((2x - 11)\)
• The second binomial: \((6x + 1)\)

When we apply the distributive property, we multiply:

• First terms: \((2x) \times (6x) = 12x^2\)
• Outer terms: \((2x) \times (1) = 2x\)
• Inner terms: \((-11) \times (6x) = -66x\)
• Last terms: \((-11) \times (1) = -11\)

Understanding how to apply this property correctly ensures accurate results in polynomial multiplication.

Combining Like Terms

Once you've applied the distributive property, the next step is to simplify the expression by combining like terms. In algebra, like terms are terms that have the same variables raised to the same power.
For instance, terms involving the variable \(x\) in our example are like terms. To combine them,
add or subtract their coefficients. In our problem,

• We combine \(2x\) and \(-66x\):
  • \(2x - 66x = -64x\)

Thus, the simplified expression becomes \(12x^2 - 64x - 11\). Combining like terms makes complex expressions easier to understand and solve.

Binomials

A binomial is a polynomial with two terms. In algebra, we often deal with the multiplication of binomials as it forms the basis of many operations.
The binomials in our exercise are \((2x - 11)\) and \((6x + 1)\). When multiplying these two binomials,
we utilize the distributive property to ensure all terms are accounted for. This results in an expression that initially has more than four terms due to each term in the first binomial being multiplied by each term in the second.
It's important to accurately distribute each term during initial multiplication. Understanding binomials lays down the foundation for more advanced algebraic concepts.

Algebra Step-by-Step

Successfully tackling algebra problems often necessitates following a logical, step-by-step process. This systematic approach ensures that each part of an expression is addressed and simplifies the problem into understandable pieces.
In our given exercise of multiplying binomials, we begin with a clear plan:

• Apply the distributive property.
• Multiply each term from one binomial with every term from the other.
• Combine like terms to simplify the expression.
• Write down the final expression.

Each step is crucial to reaching the correct answer, helping break down what might initially seem like a complex problem into manageable parts. Mastering this step-by-step methodology boosts overall problem-solving skills in algebra.