Problem 127 Simplify. $$\left(3 a^{2}-4 a+2\... [FREE SOLUTION]

Chapter 9: Problem 127

Simplify. $$\left(3 a^{2}-4 a+2\right)^{2}$$

Short Answer

Expert verified

The simplified form of $(3a^2 - 4a + 2)^2$ is $9a^4 + 24a^3 + 28a^2 - 16a + 4$.

Step by step solution

Write out the basic structure

Basically, the formula for the square of a trinomial like $(a - b + c)^2$ is $a^2 - 2ab + 2ac + b^2 - 2bc + c^2$. So, we'll structure the given trinomial in this format: $(3a^2)^2 - 2*(3a^2)*(-4a) + 2*(3a^2)*2 + (-4a)^2 - 2*(-4a)*2 + 2^2$.

Perform the operations

Now, calculate the above expression: $9a^4 + 24a^3 + 12a^2 + 16a^2 - 16a + 4$.

Combine like terms

To simplify further, combine terms with the same powers of $a$, resulting in: $9a^4 + 24a^3 + 28a^2 - 16a + 4$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trinomial Expansion

Trinomial expansion is a mathematical technique used to expand algebraic expressions with three different terms raised to a power, often squared or cubed. In this context, it involves expanding the square of a trinomial expression, such as $(3a^2 - 4a + 2)^2$. When approaching trinomial expansion:

The goal is to apply the square expansion formula: $(a - b + c)^2 = a^2 - 2ab + 2ac + b^2 - 2bc + c^2$.
Each term in the trinomial interacts with every other term according to the formula, creating multiple products that need to be calculated.
This method allows us to break complex expressions into simpler components that are easier to handle.

Understanding trinomial expansion helps in systematically breaking down and simplifying expressions that otherwise seem complex, making it easier to manage the algebraic operations ahead.

Polynomial Simplification

Polynomial simplification involves reducing an expanded polynomial into its simplest form. This process often comes after expanding expressions like the trinomial discussed earlier. During polynomial simplification, follow these key steps:

Identify and perform all the multiplications and basic arithmetic computations defined by the expansion process.
Ensure that each term is calculated accurately, keeping powers and coefficients in check.
The result might look extensive initially, but it's important to ensure that all components are expressed before combining.

By systematically simplifying the polynomial, we arrive at a neat expression that is clear and easier to work with in further analysis or problem-solving. The process reveals the essential terms needed for future calculations.

Combining Like Terms

Combining like terms is a crucial step in fully simplifying any polynomial expression. This step involves adding or subtracting terms that have identical variables raised to the same power:

First, identify terms that share the same variable and exponent. For instance, identify all terms involving $a^2$ from the expanded expression.
Add or subtract these terms' coefficients to condense them into a single term.
Repeat the process for each group of similar terms, such as $a^3$, $a^2$, $a$, and constant terms.

By combining like terms, the expression becomes more concise, which not only saves space but also simplifies subsequent computational or algebraic processes. This brings clarity and helps in better understanding the relationships between terms in the polynomial equation.

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91影视

Simplify. $$\left(3 a^{2}-4 a+2\right)^{2}$$

Short Answer

Step by step solution

Write out the basic structure

Perform the operations

Combine like terms

Key Concepts

Trinomial Expansion

Polynomial Simplification

Combining Like Terms

One App. One Place for Learning.

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