Problem 46 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(3x^5 + 7x^4 - x^3 - 3x^2 - 4x + 2\).

Step 1: Apply the Distributive Property

To multiply two polynomials, use the distributive property. Multiply each term in the first polynomial by each term in the second polynomial. Start with the first term in the first polynomial: \(3x^3(x^2 + 2x - 1)\). Distribute and multiply: \(3x^3 \cdot x^2 = 3x^5\), \(3x^3 \cdot 2x = 6x^4\), \(3x^3 \cdot (-1) = -3x^3\).

Step 2: Continue Distributing Other Terms

Now multiply the second term of the first polynomial by each term of the second polynomial: \(x^2(x^2 + 2x - 1)\). Distribute and calculate: \(x^2 \cdot x^2 = x^4\), \(x^2 \cdot 2x = 2x^3\), \(x^2 \cdot (-1) = -x^2\).

Step 3: Distribute the Last Term

Finally, multiply the last term of the first polynomial by each term of the second polynomial: \((-2)(x^2 + 2x - 1)\). Distribute and calculate: \(-2 \cdot x^2 = -2x^2\), \(-2 \cdot 2x = -4x\), \(-2 \cdot (-1) = 2\).

Step 4: Write Combined Expression

Combine all the results from previous steps: \(3x^5 + 6x^4 - 3x^3 + x^4 + 2x^3 - x^2 - 2x^2 - 4x + 2\).

Step 5: Simplify the Expression

Combine like terms from the combined expression: \(3x^5 + 6x^4 + x^4 - 3x^3 + 2x^3 - x^2 - 2x^2 - 4x + 2 = 3x^5 + 7x^4 - x^3 - 3x^2 - 4x + 2\).

Key Concepts

Distributive Property

The Distributive Property is a crucial tool when dealing with polynomial multiplication. This property allows us to multiply a single term by each term within a parenthesis. When we multiply two polynomials, we apply this property multiple times. Each term of the first polynomial is multiplied by each term of the second polynomial.
Here’s how it works step by step:

• Multiply the first term of the first polynomial by each term in the second polynomial.
• Do the same for every other term in the first polynomial.

By methodically applying these steps, you ensure every combination of terms is accounted for, such as in our given expression \[(3x^3 + x^2 - 2)(x^2 + 2x - 1).\] This systematic approach prevents errors and guarantees that you’ve distributed each term correctly.
The result will be a much longer polynomial comprised of all possible product combinations.

Simplifying Polynomials

Simplifying polynomials involves reducing a polynomial to its simplest form. After distributing, you often end up with a very complex expression. Simplifying makes it easier to handle and understand. It involves organizing and combining terms to condense the expression.
At first, the polynomial might seem overwhelming with many terms, but simplification follows logical steps:

• Write down all the terms clearly.
• Look for terms that can be combined or simplified.
• Simplify fractions or coefficients, if necessary.

Once you have applied the distributive property to the polynomials in the example, you initially get a bunch of mixed terms: \[3x^5 + 6x^4 - 3x^3 + x^4 + 2x^3 - x^2 - 2x^2 - 4x + 2.\]
The aim is to make this expression as simple as possible by organizing and consolidating terms.

Combining Like Terms

Combining Like Terms is a fundamental concept when simplifying polynomials. Like terms have the same variables raised to the same powers. For example, terms like \[3x^4\] and \[x^4\] are like terms because they both contain the variable \[x\] raised to the fourth power.
Here are steps to combine like terms effectively:

• Identify terms that have the same variable part.
• Add or subtract the coefficients of these terms.
• Organize the expression from the highest power to the lowest power for clarity.

In our exercise example, after applying the distributive property, we consolidate terms like \[6x^4 + x^4\] and \[-3x^3 + 2x^3\] by adding their coefficients, resulting in a simplified expression: \[3x^5 + 7x^4 - x^3 - 3x^2 - 4x + 2.\] This simplification is crucial for solving equations or further manipulating expressions, making it a vital skill in algebra.