Problem 33

Question

Simplify $x(2 x-4 y)-2 x(4 x+y)$

Step-by-Step Solution

Verified

Answer

The simplified expression is $-6x^2 - 6xy$.

1Step 1: Distribute Each Term

Start by distributing the variable $x$ in the expression $x(2x - 4y)$. This means multiplying $x$ with each term within the parentheses. Similarly, distribute $-2x$ in the expression $-2x(4x + y)$.\[x(2x - 4y) = 2x^2 - 4xy\]\[-2x(4x + y) = -8x^2 - 2xy\]

2Step 2: Combine the Distributed Terms

Write down the expanded terms from Step 1 together as a single expression.\[2x^2 - 4xy - 8x^2 - 2xy\]

3Step 3: Group Like Terms

Identify and group the like terms from the expanded expression. The like terms are $2x^2$ and $-8x^2$, as well as $-4xy$ and $-2xy$.

4Step 4: Simplify the Expression

Simplify the expression by adding the coefficients of the like terms.\[2x^2 - 8x^2 = -6x^2\]\[-4xy - 2xy = -6xy\]

5Step 5: Write the Final Simplified Expression

Combine the simplified terms from Step 4. The final expression is:\[-6x^2 - 6xy\]

Key Concepts

Distributive PropertyCombining Like TermsPolynomial ExpansionSimplifying Expressions

Distributive Property

The distributive property is a fundamental rule in algebra that allows you to multiply a single term by each term in a parenthesis. It's often written as:

For any terms; a(b + c) = ab + ac

This property is crucial when working with expressions that involve binomials or polynomials and helps in breaking down complex expressions into simpler parts for easier manipulation.

In our exercise, we applied the distributive property as follows:

For the expression $x(2x - 4y)$, we distributed $x$ to both $2x$ and $-4y$, which resulted in $2x^2 - 4xy$.
Similarly, for the expression $-2x(4x + y)$, we distributed $-2x$ giving us $-8x^2 - 2xy$.

Understanding how to correctly apply the distributive property can help simplify complex algebraic expressions by breaking them down into smaller, more manageable pieces.

Combining Like Terms

Combining like terms is an essential skill in algebra that further simplifies an expression after applying the distributive property. Like terms are those that include the same variables raised to the same power.

For instance:

Terms like $3x^2$ and $-5x^2$ are like terms because they both include $x^2$.
Terms like $4xy$ and $-2xy$ are also like terms since both contain the same variables $x$ and $y$.

Combining them involves adding or subtracting the coefficients, keeping the variable part unchanged. In our problem:

We combined $2x^2$ and $-8x^2$ into $-6x^2$.
We combined $-4xy$ and $-2xy$ into $-6xy$.

Polynomial Expansion

Polynomial expansion involves writing a polynomial in a longer form when terms are multiplied out. This process often uses the distributive property. It's an essential skill for translating expressions involving parentheses into summed terms.

Consider a simple binomial example:

$(x+2)(x+3)$ expands to $x(x+3)+2(x+3)$.
Finally, this becomes $x^2 + 3x + 2x + 6$ upon applying distributive property to each term.

In the original exercise, we expanded:

$x(2x-4y)$ into $2x^2 - 4xy$
$-2x(4x+y)$ into $-8x^2 - 2xy$

This breakup into individual terms lets you easily simplify by combining like terms later.

Simplifying Expressions

Simplifying expressions is the process of reducing an expression into its most concise form, making them easier to work with. After performing polynomial expansion and combining like terms, simplifying is the final step. It doesn't change the value of the expression but makes it cleaner.

When simplifying: