Problem 47 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ \left(x^{2}+x-2\right)\left(x^{3}-x+1\right) $$

Step-by-Step Solution

Answer

\(x^5 + x^4 - 3x^3 + 3x - 2\)

Step 1: Distribute Each Term

Let's start by distributing each term in the first polynomial \(x^2 + x - 2\) to each term in the second polynomial \(x^3 - x + 1\). This means multiplying every term in the first polynomial by every term in the second polynomial.

Step 2: Multiply Terms

Multiply each term in \(x^2 + x - 2\) by each term in \(x^3 - x + 1\):- \(x^2 \cdot x^3 = x^5\)- \(x^2 \cdot (-x) = -x^3\)- \(x^2 \cdot 1 = x^2\)- \(x \cdot x^3 = x^4\)- \(x \cdot (-x) = -x^2\)- \(x \cdot 1 = x\)- \(-2 \cdot x^3 = -2x^3\)- \(-2 \cdot (-x) = 2x\)- \(-2 \cdot 1 = -2\)

Step 3: Combine Like Terms

Now combine all the terms obtained:\[x^5 + x^4 - x^3 - 2x^3 + x^2 - x^2 + x + 2x - 2\]Combine the like terms:- There is only one \(x^5\) term.- Combine \(x^4\) terms: \(x^4\)- Combine \(x^3\) terms: \(-x^3 - 2x^3 = -3x^3\)- Combine \(x^2\) terms: \(x^2 - x^2 = 0\)- Combine \(x\) terms: \(x + 2x = 3x\)- \(-2\) remains as it is.

Step 4: Write the Simplified Expression

The simplified form of the expression after combining like terms is:\[x^5 + x^4 - 3x^3 + 3x - 2\]

Key Concepts

Distributive Property

Polynomial multiplication heavily relies on the distributive property. This property states that a term in one expression should be multiplied by each term within another expression. When you multiply polynomials like \((x^2 + x - 2)(x^3 - x + 1)\), apply the distributive property by taking each term from the first polynomial and multiplying it with every term in the second polynomial.

Here's how:

• Multiply the first term in the first polynomial by every term in the second polynomial.
• Repeat this for the second term of the first polynomial, and so on.

This ensures that all possible combinations of terms are multiplied together. It might seem like a lot of steps, but handling them one by one makes it manageable.
By using the distributive property methodically, you ensure every interaction between terms is accounted for in polynomial multiplication.

Combining Like Terms

After using the distributive property, you'll end up with a set of polynomial terms. Many of these will be alike and should be combined to simplify the expression. Like terms have the same variable raised to the same power. For example, consider \(-x^3 - 2x^3\). Both of these are "like terms" because each has the variable "x" raised to the power of 3.

To combine them, add or subtract the coefficients:

• In \(-x^3 - 2x^3\), combine by adding these coefficients to get \(-3x^3\).
• Similarly, \(x + 2x = 3x\) by adding the coefficients (1x and 2x).

Combining like terms helps simplify the expression by reducing the number of terms and making calculation or further simplification easier.

Simplifying Polynomials

Once you have combined the like terms, your polynomial is almost fully simplified. Simplifying polynomials means writing them in the most condensed form possible without losing any necessary details. After combining, your expression \(x^5 + x^4 - 3x^3 + 3x - 2\) contains no like terms left to combine.

Steps to ensure it's simplified:

• Check that each term has a distinct exponent.
• Ensure all coefficients are as simplified as possible.
• The polynomial terms are typically written in descending order by degree, which in this case is the case: starting from \(x^5\) and ending with the constant (-2).

Simplifying allows you to express the polynomial in its "cleanest" form, making it easier to use in further calculations or applications.