Problem 55 - Algebra A Combined Function

Question

Multiply vertically. \((x+3)\left(2 x^{2}+4 x-1\right)\)

Step-by-Step Solution

Answer

The product is \(2x^3 + 10x^2 + 11x - 3\).

Step 1: Distribute Each Term Separately

First, distribute the terms in the binomial \(x+3\) across the trinomial term by term. This means you will multiply each term in \(x+3\) by each term in \(2x^2 + 4x - 1\) separately.

Step 2: Multiply Each Pair of Terms

Perform the multiplication for each combination.1. Multiply \(x\) by each term in \(2x^2 + 4x - 1\): - \(x imes 2x^2 = 2x^3\) - \(x imes 4x = 4x^2\) - \(x imes -1 = -x\)2. Multiply \(3\) by each term in \(2x^2 + 4x - 1\): - \(3 imes 2x^2 = 6x^2\) - \(3 imes 4x = 12x\) - \(3 imes -1 = -3\)

Step 3: Combine Like Terms

Now, add together all the results from Step 2, ensuring to combine like terms:- \(2x^3 + (4x^2 + 6x^2) + (-x + 12x) - 3\)- Combine like terms: \rightarrow 2x^3 + 10x^2 + 11x - 3\.

Key Concepts

Using the Distributive Property

The distributive property is a fundamental aspect of multiplying polynomials. This property essentially states that every single part of one polynomial must be multiplied by every single part of another polynomial, ensuring that all components are accounted for. In this exercise, the binomial \(x+3\) is multiplied with the trinomial \(2x^2 + 4x - 1\). To effectively utilize the distributive property, follow these steps:

• Take the first term from the binomial (in this case, \(x\)).
• Multiply it by each term within the trinomial: \(2x^2\), \(4x\), and \(-1\).
• Next, take the second term from the binomial (3), and multiply it by each term within the trinomial as well.

This multiplication ensures that each potential combination of terms is considered. Using the distributive property is essential because it provides a methodical way to handle polynomial multiplication and avoids missing any terms.

Combining Like Terms

Combining like terms is the necessary step to simplify expressions after multiplying each term. It involves adding together terms that have the same variables raised to the same power. To simplify, follow these steps:

• Look at the resulting terms from the multiplication process: \(2x^3\), \(4x^2\), \(6x^2\), \(-x\), \(12x\), and \(-3\).
• Identify and group together like terms. Recall that 'like terms' are terms with exactly the same variable part (e.g., all \(x^2\) terms).
• Combine these terms by adding or subtracting their coefficients. For example, add \(4x^2\) and \(6x^2\) to get \(10x^2\), and \(-x\) with \(12x\) to get \(11x\).

The final expression, in its simplest form, \(2x^3 + 10x^2 + 11x - 3\), is achieved by carefully combining all like terms. It’s vital to do so to ensure clarity and accuracy in mathematical expressions.

Binomial and Trinomial Multiplication

Binomial and trinomial multiplication often seems daunting at first, but understanding each step simplifies the process dramatically. A binomial has two terms, such as \(x + 3\), while a trinomial has three, like \(2x^2 + 4x - 1\). The objective is to multiply these polynomials together, resulting in a polynomial with potentially more terms.

• Start by applying the distributive property: multiply each term in the binomial by every term in the trinomial.
• This approach involves creating pairs for multiplication, which results in terms like \(2x^3\), \(4x^2\), \(-x\), \(6x^2\), \(12x\), and \(-3\).
• After obtaining these individual products, remember to combine the like terms to consolidate the expression into its simplest form.

By breaking down the multiplication process into smaller steps and recognizing patterns in like terms, polynomial multiplication becomes an easier task. This technique is useful and widely applicable in many areas of mathematics.