Problem 44 - Algebra 2

Question

Simplify. $$ \frac{3}{4} x^{2}\left(8 x+12 y-16 x y^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( 6x^3 + 9x^2y - 12x^3y^2 \).

Step 1: Distribute the term

Let's distribute the fraction \( \frac{3}{4}x^2 \) to each term inside the parentheses: \( \frac{3}{4}x^2(8x) + \frac{3}{4}x^2(12y) - \frac{3}{4}x^2(16xy^2) \).

Step 2: Simplify each term

Simplify each term from the distribution. For \( \frac{3}{4}x^2(8x) \):- Multiply coefficients: \( \frac{3}{4} \times 8 = 6 \)- Combine powers of \( x \): \( x^2 \times x = x^3 \)- Result: \( 6x^3 \)For \( \frac{3}{4}x^2(12y) \):- Multiply coefficients: \( \frac{3}{4} \times 12 = 9 \)- Combine powers: \( x^2 \times y = x^2y \)- Result: \( 9x^2y \)For \( \frac{3}{4}x^2(16xy^2) \):- Multiply coefficients: \( \frac{3}{4} \times 16 = 12 \)- Combine powers: \( x^2 \times x = x^3 \) and \( y^2 \)- Result: \( 12x^3y^2 \)

Step 3: Combine simplified expressions

After simplifying each term, combine the results: \[ 6x^3 + 9x^2y - 12x^3y^2 \]. This is the simplified form of the given expression.

Key Concepts

Distributive Property

The distributive property is a crucial concept in algebra that allows us to multiply a single term by multiple terms inside parentheses. It looks like this:

• If you have an expression such as \(a(b + c)\), you can distribute \(a\) across \(b\) and \(c\). This becomes \(ab + ac\).

Let's consider the exercise:

• We have the term \( \frac{3}{4}x^2 \) that needs to be multiplied by each of the terms inside the parentheses \((8x + 12y - 16xy^2)\).

Using the distributive property, we obtain:

• \( \frac{3}{4}x^2(8x) + \frac{3}{4}x^2(12y) - \frac{3}{4}x^2(16xy^2) \).

Combining Like Terms

Combining like terms is a technique used to simplify expressions by merging terms that have the same variables raised to the same power.

• In our simplified expression \(6x^3 + 9x^2y - 12x^3y^2\), observe that each term is different from the others, indicating no further combinations are possible.

However, if there were terms like \(5x^2 + 4x^2\), you could combine them:

• Add the coefficients: \(5 + 4 = 9\) and keep the variable part \(x^2\).
• Result: \(9x^2\).
It's essential to identify terms that can be combined to simplify polynomial expressions effectively.

Exponents

Exponents are shorthand for repeated multiplication of the same base. When simplifying expressions, you often need to manage exponents accurately.

• For example, in the term \(x^2 \times x\), you add the exponents of the same base, resulting in \(x^3\).
• For another instance: In the expression \(x^2 \times y\), the exponents only apply to their respective bases, making the combined form \(x^2y\).

Remember:

• Addition of exponents happens when you multiply like bases.
• Keep the base and add the powers together.

Exponents make it easier to express and manipulate large numbers or repeated multiplications in polynomial expressions.

Polynomial Expressions

Polynomial expressions consist of variables raised to whole number exponents and their coefficients. They can include multiple terms added or subtracted from each other.

• Examples include \(x^2 + 3x + 2\) or \(6x^3 + 9x^2y - 12x^3y^2\).

In the exercise you worked on, you dealt with a polynomial expression involving multiple terms inside parentheses, which got distributed and then simplified.

• To further understand polynomial expressions, remember that each component within a polynomial must be a non-negative integer power of the variables.

In more complex operations, you may encounter the need to add, subtract, or even multiply polynomials, where understanding each term and its degree becomes crucial.