Problem 77 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$\left(x^{2}+2\right)(3 x-2)$$

Step-by-Step Solution

Answer

The product is \(3x^3 - 2x^2 + 6x - 4\).

Step 1: Distribute the First Term

Start by distributing the first term of the first polynomial, which is \(x^2\), across each term in the second polynomial: \(3x\) and \(-2\). This gives:\[x^2 \cdot 3x = 3x^3\] \[x^2 \cdot (-2) = -2x^2\]

Step 2: Distribute the Second Term

Now distribute the second term of the first polynomial, which is \(2\), across each term in the second polynomial: \(3x\) and \(-2\). This results in:\[2 \cdot 3x = 6x\] \[2 \cdot (-2) = -4\]

Step 3: Combine the Results

Add all the products obtained from Steps 1 and 2 together to get the resulting polynomial:\[3x^3 - 2x^2 + 6x - 4\]

Key Concepts

Distributive Property

The distributive property is a crucial tool in algebra that allows us to simplify and multiply expressions. It states that for any numbers or expressions, say \(a\), \(b\), and \(c\), the expression \(a(b+c)\) can be distributed as \(ab + ac\). This principle is not only vital in basic arithmetic but also in handling complex polynomial expressions. In the given exercise, we see this property at work when multiplying the first polynomial \((x^2 + 2)\) by each term of the second polynomial \((3x - 2)\). By distributing, each term from the first polynomial is multiplied individually with the terms from the second polynomial. This step-wise distribution is a reliable method to ensure you don't miss any products.

Polynomial Expressions

Polynomial expressions are mathematical phrases featuring variables raised to whole number powers and combined using addition, subtraction, or multiplication. They can range from simple expressions like \(x + 2\) to more complex forms like \(3x^3 - 2x^2 + 6x - 4\). In our example, we are dealing with two polynomial expressions, \(x^2 + 2\) and \(3x - 2\). When we multiply these expressions, we are essentially finding their product through distributing and combining similar terms. Remember, a polynomial's degree is determined by its highest power of the variable, so upon multiplying, new terms (and thus a new degree) emerge as seen in the result \(3x^3 - 2x^2 + 6x - 4\) with the highest degree being 3.

Algebraic Manipulation

Algebraic manipulation involves rearranging expressions and equations to simplify or solve them. This skill is key when working with polynomials, especially in multiplication. In our problem, algebraic manipulation helps us to systematically distribute each term and combine like terms to simplify the expression. It involves several steps, including:

• Applying the distributive property to ensure every term is multiplied correctly.
• Simplifying each individual product before combining them.
• Finally, summing up all the products and merging like terms for a neat and concise polynomial expression like \(3x^3 - 2x^2 + 6x - 4\).

This process not only simplifies the expression but also enhances our understanding of how each part of the expression contributes to the overall polynomial. By mastering algebraic manipulation, students can tackle a wide range of polynomial problems with confidence.