Problem 65 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ -5 x^{2}(x+2) $$

Step-by-Step Solution

Answer

Question: Simplify the expression: \(-5x^2(x+2)\). Answer: The simplified expression is \(-5x^3-10x^2\).

Step 1: Distribute the \(-5x^2\) term

Multiply \(-5x^2\) by each term in the parentheses \((x+2)\). This will give us a new expression: $$ -5x^2(x)+(-5x^2)(2) $$

Step 2: Multiply the terms

Now, we need to multiply the terms obtained in the previous step. Multiply \(-5x^2\) by \(x\) and \(-5x^2\) by \(2\). We get: $$ -5x^3-10x^2 $$

Step 3: Look for like terms to combine

In this expression, there are no like terms to combine. So, the simplified expression is: $$ -5x^3-10x^2 $$ The simplified expression after performing the multiplications and combining like terms is \(-5x^3-10x^2\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across a sum or difference within parentheses. Essentially, this property tells us that if you have a scenario like \( a(b + c) \), it's equivalent to \( ab + ac \).

Let's have a closer look at the problem \( -5x^2(x + 2) \). Here, we apply the distributive property by multiplying \( -5x^2 \) with each term inside the parentheses - that is, \( x \) and \( +2 \). By doing so, we get \( -5x^2 \times x + (-5x^2) \times 2 \), which leads to the new expression \( -5x^3 - 10x^2 \).

Crucially, when applying the distributive property, be careful with signs; multiplying by a negative term, such as \( -5x^2 \), impacts the sign of each term you're distributing across.

Combining Like Terms

Once the distributive property has been applied and the multiplication is performed, the next step is often to simplify the expression by combining like terms. Like terms are terms that have the exact same variable parts, meaning they have the same variables raised to the same powers. For instance, \( 3x^2 \) and \( -7x^2 \) are like terms, while \( x^2 \) and \( x^3 \) are not.

In the example \( -5x^3 - 10x^2 \), we do not have like terms since one term is an \( x^3 \) term and the other is an \( x^2 \) term. Had we had a situation with terms like \( -5x^3 + 3x^3 \), we would combine them to get \( -2x^3 \), simplifying the polynomial.

Remember, the goal of combining like terms is to simplify the algebraic expression as much as possible, which can make further algebraic manipulations and solving equations far more manageable.

Polynomial Multiplication

Polynomial multiplication might seem daunting at first, but by applying the distributive property along with combining like terms, it becomes a systematic process. A polynomial is an algebraic expression consisting of terms with variables raised to whole number exponents and their coefficients. When multiplying polynomials, you'll want to distribute each term of one polynomial across each term of the other.

For instance, if you're multiplying \( (3x + 2) \times (x^2 + x + 1) \), you'd multiply 3x by each term in the second polynomial and 2 by each term in the second polynomial, then combine like terms at the end. The key is to be methodical, multiply each term carefully, and then simplify.

In the step-by-step solution provided, \( -5x^2(x + 2) \) is a case of multiplying a monomial by a binomial, which is simpler than the example just provided but follows the same principles. By understanding these concepts, you're well on your way to mastering polynomials and algebra in general.