Problem 20 - Algebra 1: Concepts and Skills

Question

$$ (3 x-4)(2 x-1) $$

Step-by-Step Solution

Answer

The product of given binomial expressions is \(6x^2 - 11x + 4\)

Step 1: Distribute the First terms

Multiplying the first terms of each binomials together, we get \(3x \cdot 2x = 6x^2\)

Step 2: Distribute the Outer terms

Multiplying the outer terms of the expression, we get \(3x \cdot -1 = -3x\)

Step 3: Distribute the Inner terms

Multiplying the inner terms of the expression, we get \(-4 \cdot 2x = -8x\)

Step 4: Distribute the Last terms

Multiplying the last terms of the expression, we get \(-4 \cdot -1 = 4\)

Step 5: Add all terms

Finally, we sum all the computed expressions. Our final answer should be \(6x^2 - 3x - 8x + 4\). The result can be simplified by combining like terms.

Step 6: Combine Like Terms

Upon combining like terms, we have \(6x^2 - 11x + 4\). This is our final answer.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across a binomial or other expression. It states that for any integers or expressions, the multiplication over addition or subtraction is given by:

• \( a(b + c) = ab + ac \)
• This can also be extended to subtraction: \( a(b - c) = ab - ac \)

In our exercise, we applied the distributive property to multiply two binomials: \((3x-4)\) and \((2x-1)\). We executed this by separately multiplying each term of the first binomial by each term in the second binomial.
By following the steps outlined, we multiplied the first term of the first binomial with the first term of the second, termed as 'distributing the first terms', and did similarly for all possible combinations. This application is critical in polynomial multiplication for ensuring each term is methodically accounted for.

Binomials

A binomial is a polynomial with exactly two terms, typically connected by a plus or minus sign. In our example, both expressions \((3x-4)\) and \((2x-1)\) are binomials.
Binomials are an essential building block in algebra. Each term in a binomial can be a number, a variable, or a product of numbers and variables.

• The binomial \((3x-4)\): Here, '3x' and '-4' are the terms.
• The binomial \((2x-1)\): Contains the terms '2x' and '-1'.

When multiplying binomials, we use a method that's often remembered by the acronym FOIL (First, Outer, Inner, Last). This helps in systematically distributing terms. The order in FOIL corresponds to the order we multiply the terms in to apply the distributive property effectively. Understanding binomials is foundational for simplifying complex algebraic expressions and equations.

Combining Like Terms

Combining like terms is an essential process in simplifying algebraic expressions. Like terms are terms in an expression that have the same variable(s) and power(s). By combining these terms, we create a simpler and more concise expression.
In the final step of solving the given expression, we identified like terms - which were the terms containing '\(x\)'. From the expression \(6x^2 - 3x - 8x + 4\), \(-3x\) and \(-8x\) are like terms. So, we added them together to get \(-11x\).

• This simplification led us to the final result: \(6x^2 - 11x + 4\).

By combining like terms, the expression not only becomes simpler but also cleaner, making it easier to interpret and work with further. Understanding and applying this concept enhances algebraic manipulation skills.