Problem 83 - Algebra 1

Question

Find the product. $$(6 x+2)\left(x^{2}-x-1\right)$$

Step-by-Step Solution

Answer

The product of \((6x+2)(x^{2}-x-1)\) is \(6x^{3}-4x^{2}-8x-2\)

Step 1: Distribute First Term

First, distribute the first term of the binomial \(6x\) to each term in the trinomial: \(6x*x^{2}\), \(6x*-x\), and \(6x*-1\). This gives: \(6x^{3}-6x^{2}-6x.\)

Step 2: Distribute Second Term

Next, distribute the second term of the binomial \(2\) to each term in the trinomial: \(2*x^{2}\), \(2*-x\), and \(2*-1\). This gives: \(2x^{2}-2x-2.\)

Step 3: Combine like terms

The final step is to add the two results together. Combine the terms from Step 1 and Step 2 to get the final result. This yields: \(6x^{3}-6x^{2}-6x+2x^{2}-2x-2 = 6x^{3}-4x^{2}-8x-2.\)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra which allows us to simplify expressions and makes polynomial multiplication manageable. When you apply this property, you multiply each term of one polynomial by every term of another polynomial. Here, the process happens by distributing terms across an expression.

For our exercise, we have a binomial \(6x + 2\) and a trinomial \(x^2 - x - 1\). According to the distributive property, we start by distributing each term of the binomial to every term in the trinomial:

• First, distribute \(6x\): Multiply \(6x\) by each term in the trinomial \(x^2 - x - 1\). This results in \(6x(x^2) - 6x(x) - 6x(1)\), which simplifies to \(6x^3 - 6x^2 - 6x\).
• Next, distribute the constant term \(2\): Multiply \(2\) by each term of the trinomial to get \(2(x^2) - 2(x) - 2(1)\), yielding \(2x^2 - 2x - 2\).

By recapping the distributive property, you ensure that every term is accounted for and multiplied correctly, setting the stage for combining like terms in the next steps.

Combining Like Terms

Combining like terms is the next step after applying the distributive property. This involves simplifying the expression by adding or subtracting terms that have identical variable parts. In polynomial operations, it’s crucial to keep things tidy and concise by merging these alike terms.

From our distributed terms, we have two groups of terms to act upon:

• The expression from step 1: \(6x^3 - 6x^2 - 6x\)
• The expression from step 2: \(2x^2 - 2x - 2\)

To combine them, simply add together the coefficients of terms that share the same variable and power:

• Add \(6x^3\) from step 1 with no corresponding term from step 2, so it remains as \(6x^3\).
• Combine \(-6x^2\) with \(2x^2\) to get \(-4x^2\).
• Combine \(-6x\) with \(-2x\) to get \(-8x\).
• The constant term is simply \(-2\) since there is no other constant to combine it with.

The combined expression, now simpler, is \(6x^3 - 4x^2 - 8x - 2\), making it much easier to understand and work with.

Binomial and Trinomial

In polynomial multiplication, understanding the types of polynomials you are working with is essential. A binomial is a polynomial with two terms, while a trinomial has three. They are fundamental building blocks in algebra.

In this exercise, the binomial \(6x + 2\) interacts with the trinomial \(x^2 - x - 1\). By knowing what each term and type of polynomial is, you can strategically apply operations like multiplication.

• A **binomial** offers a simple yet pivotal structure for operations like distribution, making it achievable to spread each term across another polynomial.
• A **trinomial** allows for richer algebraic manipulation, offering three terms that can blend with the terms from a binomial to create more extensive expressions.

By comprehending these basic structures, you gain confidence in performing polynomial multiplication, learning that even complex expressions can be broken down by applying known operations methodically.