Problem 12 In exercises $6-12,$ expand an... [FREE SOLUTION]

Chapter 1: Problem 12

In exercises $6-12,$ expand and simplify the given expressions. $$ 3\left(\theta^{2}+4\right)^{2}(2 \theta) $$

Short Answer

Expert verified

The expression simplifies to $6\theta^5 + 48\theta^3 + 96\theta$.

Step by step solution

Rewrite the Expression

The given expression is $ 3(\theta^2 + 4)^2 (2\theta) $. To make it easier to handle, we'll first rewrite it to see more clearly how all factors are combined: $ 3 \times (\theta^2 + 4)^2 \times (2\theta) $.

Simplify by Distributing the Constants

First, we'll distribute the constants. Since the expression consists of $ 3 \times 2\theta $, we multiply the constants $3$ and $2$ together: $6\theta \times (\theta^2 + 4)^2 $.

Expand the Squared Term

Now, we need to expand $(\theta^2 + 4)^2$. This involves algebraic expansion: $ (\theta^2 + 4) \times (\theta^2 + 4) = \theta^4 + 8\theta^2 + 16 $.

Distribute $6\theta$ Across the Expanded Terms

Using the expanded form $\theta^4 + 8\theta^2 + 16$, distribute $6\theta$ across each term: $6\theta \times \theta^4 = 6\theta^5$$6\theta \times 8\theta^2 = 48\theta^3$$6\theta \times 16 = 96\theta$

Combine the Terms into a Single Expression

Combine the distributed terms into one expression to get the final simplified form: $6\theta^5 + 48\theta^3 + 96\theta$.

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Key Concepts

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

Polynomial Simplification

Polynomial simplification is a process used to transform a complicated polynomial expression into a simpler form. This involves removing any redundant terms, combining like terms, and arranging terms in decreasing order of degrees.

Simplification often helps in improving readability and understanding, especially in solving equations.
It's also crucial when you need to substitute values or further manipulate the polynomial.

In our original problem, after expanding and distributing correctly, we finally arrive at a simplified polynomial: $ 6\theta^5 + 48\theta^3 + 96\theta $. Here, the terms are neatly combined into a format that鈥檚 easy to interpret.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions by eliminating parentheses. It states that for all numbers or variables $a, b,$ and $c$, the formula $a(b + c) = ab + ac$ holds true.

This allows multiplication over addition or subtraction to be executed seamlessly.
It is particularly useful in expanding expressions and solving equations efficiently.

In the context of the exercise given here, we initially distribute a constant over a polynomial expression using this property: $3 \times (2\theta) = 6\theta$. Later, it helps in distributing $6\theta$ across each term of the expanded polynomial $(\theta^4 + 8\theta^2 + 16)$.

Exponentiation

Exponentiation refers to the process of raising a number or expression to a power or exponent, characterized by repeated multiplication. For example, $x^n$ means multiplying $x$ by itself $n$ times.

It can greatly simplify expressions that contain repeated factors.
Understanding exponent laws is essential in many fields of mathematics.

In this problem, we dealt with exponentiation during the expansion of $(\theta^2 + 4)^2$. This required multiplying $(\theta^2 + 4)$ by itself, resulting in the expanded form: $\theta^4 + 8\theta^2 + 16$. Here, exponentiation helped introduce higher-degree terms into the polynomial.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial raised to a power as a sum of terms. It requires careful application of both distributive property and exponentiation.

Expanding polynomials helps identify each term's contribution to the overall expression.
This step often utilizes binomial expansions if the polynomial fits a binomial pattern.

In the exercise at hand, expanding $(\theta^2 + 4)^2$ shows the expansion technique: $ (\theta^2 + 4) \times (\theta^2 + 4) $. This operation results in the terms $\theta^4 + 8\theta^2 + 16$, which then allows for further manipulation of the polynomial. To effectively expand a polynomial, both multiplication skills and patience are key.

91影视

In exercises \(6-12,\) expand and simplify the given expressions. $$ 3\left(\theta^{2}+4\right)^{2}(2 \theta) $$

Short Answer

Step by step solution

Rewrite the Expression

Simplify by Distributing the Constants

Expand the Squared Term

Distribute \(6\theta\) Across the Expanded Terms

Combine the Terms into a Single Expression

Key Concepts

Polynomial Simplification

Distributive Property

Exponentiation

Polynomial Expansion

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