Problem 45 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Distribute the First Polynomial

The first step is to distribute each term in the first polynomial, \( x^2 + x - 1 \), across the second polynomial, \( 2x^2 - x + 2 \). Start by distributing \( x^2 \):\[x^2 \cdot (2x^2 - x + 2) = 2x^4 - x^3 + 2x^2\]

Step 2: Distribute the Second Term

Next, distribute the second term, \( x \), across the second polynomial:\[x \cdot (2x^2 - x + 2) = 2x^3 - x^2 + 2x\]

Step 3: Distribute the Third Term

Now distribute the third term, \( -1 \), across the second polynomial:\[-1 \cdot (2x^2 - x + 2) = -2x^2 + x - 2\]

Step 4: Combine Like Terms

Now, combine all terms from the distribution steps:\[2x^4 - x^3 + 2x^2 + 2x^3 - x^2 + 2x - 2x^2 + x - 2\]Combine like terms to simplify:- For \( x^4 \): \( 2x^4 \)- For \( x^3 \): \( -x^3 + 2x^3 = x^3 \)- For \( x^2 \): \( 2x^2 - x^2 - 2x^2 = -x^2 \)- For \( x \): \( 2x + x = 3x \)- Constant term: \( -2 \) remains as it is.

Step 5: Write the Final Expression

The simplified expression is:\[2x^4 + x^3 - x^2 + 3x - 2\]

Key Concepts

Distribution

Distribution is a method used in polynomial multiplication where each term in one polynomial is multiplied by each term in another polynomial. This approach allows us to systematically handle each term to ensure no term is left out. For example, in the polynomial multiplication \((x^2 + x - 1)(2x^2 - x + 2)\), we start by distributing the first term, \(x^2\), across all terms of the second polynomial:

• \(x^2 \cdot 2x^2 = 2x^4\)
• \(x^2 \cdot -x = -x^3\)
• \(x^2 \cdot 2 = 2x^2\)

Next, we move to the second term, \(x\), and do the same:

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

Finally, distribute the third term, \(-1\):

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

This step ensures every part of the first polynomial is considered against every part of the second.

Combining Like Terms

After distributing, you'll notice we have several terms that look similar. These are what we call 'like terms', and they are terms that have the same variable raised to the same power. To simplify our polynomial, we need to combine these like terms. In our example, the expression after distribution looks like this:\[2x^4 - x^3 + 2x^2 + 2x^3 - x^2 + 2x - 2x^2 + x - 2\] Let's group them by their degree and combine them:

• For \(x^4\): There is only one term, \(2x^4\).
• For \(x^3\): Combine \(-x^3\) and \(2x^3\) to get \(x^3\).
• For \(x^2\): Combine \(2x^2 - x^2 - 2x^2\) to get \(-x^2\).
• For \(x\): Combine \(2x + x\) to get \(3x\).
• Constant Term: \(-2\) remains unchanged.

This process of combining like terms reduces the expression to fewer terms and makes it easier to understand and use in calculations and interpretations.

Simplification

Simplification is the final step, where we write the combined terms in a neat and concise way. Each term is ordered by its degree, starting with the highest power. In our example, following distribution and combining, we arrived at:\[2x^4 + x^3 - x^2 + 3x - 2\] The polynomial is now in its simplest form, which makes it easier to understand at a glance. Simplification is important as it allows us to quickly interpret the expression and ensures that error-free future calculations can be performed efficiently without dealing with unnecessary terms. By reducing a polynomial to its simplest form, any further mathematical operations or comparisons become straightforward.In summary, simplification not only helps in reading and understanding the expression but also prepares it for any advanced operations that may be needed in subsequent problems.