Problem 73 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$(x+1)\left(x^{2}+2 x-3\right)$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are tasked with multiplying the polynomial \((x+1)\) by the polynomial \((x^2 + 2x - 3)\). This requires using the distributive property to expand the expression.

Step 2: Apply the Distributive Property

Distribute the term \(x\) from \((x+1)\) across the trinomial \((x^2 + 2x - 3)\). This gives us: \[ x(x^2 + 2x - 3) = x^3 + 2x^2 - 3x \]

Step 3: Distribute the Second Term

Next, distribute the constant term \(+1\) from \((x+1)\) across the trinomial \((x^2 + 2x - 3)\). This results in: \[ 1(x^2 + 2x - 3) = x^2 + 2x - 3 \]

Step 4: Combine Like Terms

Add the results from Step 2 and Step 3 to combine like terms. Begin by writing out:\[ x^3 + 2x^2 - 3x + x^2 + 2x - 3 \]Combine the like terms:\[ x^3 + 3x^2 - x - 3 \]

Step 5: Write the Final Result

After combining like terms, the final expanded polynomial is:\[ x^3 + 3x^2 - x - 3 \]

Key Concepts

Distributive Property

When dealing with polynomial multiplication, the distributive property is your first best friend. It helps us multiply each term inside one polynomial by every term in another. Imagine you have two polynomials, like

• \((x + 1)\)
• \((x^2 + 2x - 3)\)

You start by taking the first term of the first polynomial, which is \(x\), and distribute it across every term in the second polynomial \((x^2, 2x, \text{and} -3)\).* This results in:

• \(x \cdot x^2 = x^3\)
• \(x \cdot 2x = 2x^2\)
• \(x \cdot (-3) = -3x\)

Now, do the same with the second term, \(+1\), from the first polynomial and distribute it:

• \(1 \cdot x^2 = x^2\)
• \(1 \cdot 2x = 2x\)
• \(1 \cdot (-3) = -3\)

This way, you've expanded the product into two sets of terms, and you're ready for the next step: combining like terms.

Combine Like Terms

After using the distributive property, you'll end up with an expression full of terms, such as:\[ x^3 + 2x^2 - 3x + x^2 + 2x - 3 \]To make sense of all this, we use the concept of 'combining like terms'. Like terms are those that contain the same variables raised to the same power. The coefficients (numbers in front of the variables) can be different, but the variable part must match. Let's identify and combine them:

• The \(x^3\) term doesn't have any pairs, so it stays \(x^3\).
• Combine the \(2x^2\) and \(x^2\) to get \(3x^2\).
• The terms \(-3x\) and \(2x\) combine to \(-x\).
• And last, the constant \(-3\) is on its own.

Once combined, your expression will look cleaner and organized: \(x^3 + 3x^2 - x - 3\). By combining like terms, you simplify that big messy expression into its simplest polynomial form.

Expand Polynomials

Expanding polynomials involves taking a polynomial expression and using operations to express it in its extended form. This means multiplying the terms and using properties like the distributive property to remove any parentheses, resulting in a longer expression with individual terms presented. For example, if you have

• \((x + 1)(x^2 + 2x - 3)\)

You aim to express it without the parentheses using basic arithmetic operations. Trendy challenges include ensuring that all combinations of terms have been expanded correctly and preventing mistakes during the distribution process. By expanding polynomials, you help to identify like terms for combining and prepare the expression for further analysis, such as factoring or evaluating at given values. Expansion may initially seem complex, but practicing will help you to see patterns and become more comfortable with different polynomials.