Problem 14 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The result of the multiplication of the binomials is \(8x^2 + 14x + 3\).

Step 1: Apply the first part of FOIL

Here FOIL stands for First, Outer, Inner, Last. To apply the first part (F), multiply the first terms of both binomials: \(2x * 4x = 8x^2\).

Step 2: Apply the Outer and Inner of FOIL

Now, apply the outer (O) and inner (I) parts of FOIL. Multiply the outer terms of the binomials and then the inner terms. \(2x * 1 = 2x\) (outer) and \(3 * 4x = 12x\) (inner). Combine these results.

Step 3: Apply the Last part of FOIL

Finally, apply the last part (L) of FOIL: multiply the last terms of both binomials: \(3 * 1 = 3\).

Step 4: Combine all parts

Combine all terms we got from steps 1 to 3: \(8x^2 + 2x + 12x + 3\), which simplifies to \(8x^2 + 14x + 3\) after combining like terms.

Key Concepts

Distributive Property

When examining polynomial multiplication, one foundational concept is the Distributive Property. This mathematical principle states that multiplication distributed over addition allows you to multiply each term within the parentheses by a term outside it. This property is key when dealing with polynomials, as it provides a systematic way to multiply complex expressions.

Here's how the Distributive Property works in practice:

• The equation format is typically \(a(b + c) = ab + ac\). The term \(a\) is multiplied by each term within the parentheses.


• This same process is used to multiply binomials. For example, in the expression \((2x+3)(4x+1)\), each term in the first binomial is multiplied by each term in the second binomial separately.

By understanding this property, you gain the ability to break down polynomial expressions into simpler, more manageable parts.

Polynomial Expansion

Polynomial Expansion is the process of multiplying out and simplifying expressions like \( (2x+3)(4x+1)\) into a single polynomial. This involves distributing each term across all terms in the other polynomial. It's a methodical way to simplify complex algebraic expressions and can lead to a single, streamlined polynomial.

In the given exercise, the expansion process includes these impactful steps:

• First, multiply the individual terms to form new terms, resulting in variables raised to powers, such as \(8x^2\).


• Next, sum like terms. In this case, after expanding, you would combine terms such as \(2x\) and \(12x\) to get \(14x\).


• Finally, ensure all like terms have been appropriately added or subtracted.

By expanding a polynomial, you transform it from a product of binomials into a straightforward expression with all terms clearly defined.

FOIL Method

The FOIL Method is a valuable technique in algebra for multiplying two binomials. It's an acronym representing the steps: First, Outer, Inner, and Last. Each step involves specific pairs of terms in the binomials.

• First: Multiply the first terms in each binomial, such as \(2x\) and \(4x\), to get \(8x^2\).


• Outer: Multiply the outer terms - \(2x\) and \(1\) - yielding \(2x\).


• Inner: Next, multiply the inner terms - \(3\) and \(4x\) - which results in \(12x\).


• Last: Lastly, multiply the last terms in the binomials, \(3\) and \(1\), giving \(3\).

The importance of FOIL in binomial multiplication is its ability to systematically break down and simplify the complex task into easy, manageable stages. By completing each step, students can ensure no term is overlooked, leading to a full and accurate polynomial expansion.