Problem 62 - Algebra A Combined Function

Question

Multiply. \(-5 x\left(x^{2}-3 x+10\right)\)

Step-by-Step Solution

Answer

The result is -5x^3 + 15x^2 - 50x.

Step 1: Apply the Distributive Property

Start by distributing the term -5x to each term inside the parenthesis (x^2 - 3x + 10). The expression becomes -5x(x^2) - 5x(-3x) - 5x(10).

Step 2: Multiply Each Term

Now perform the multiplications: - -5x(x^2) = -5x^3. - -5x(-3x) = 15x^2. - -5x(10) = -50x.

Step 3: Combine the Results

Combine the results from the previous step to get the final expression: -5x^3 + 15x^2 - 50x.

Key Concepts

Distributive Property

The distributive property is a very important mathematical concept that helps in simplifying and solving algebraic expressions, especially those involving polynomials. It states that for any numbers or expressions, \( a(b + c) = ab + ac \). This means you multiply the term outside the parenthesis with each term inside it.

In the context of polynomial multiplication, like in the given exercise

• -5x(x^2 - 3x + 10),

we distribute \(-5x\) to each term inside the parenthesis.

• First, we multiply \(-5x\) with \(x^2\), resulting in \(-5x^3\).
• Next, \(-5x\) is multiplied by \(-3x\), giving us \(15x^2\).
• Finally, \(-5x\) is multiplied by 10, which results in \(-50x\).

Applying the distributive property ensures that you are considering how every term in the expression interacts with others, leading to a fully expanded form.

Monomial

A monomial is the simplest type of polynomial, consisting of only one term. It can be a constant, a variable, or a product of constants and variables raised to powers. Understanding monomials is crucial as they form the building blocks of more complex polynomial expressions.

In the exercise given, \(-5x\) is the monomial that needs to be distributed across the polynomial \(x^2 - 3x + 10\). When distributing a monomial, you simply multiply it with each term of the polynomial separately.

The key takeaway is recognizing that the degree of a monomial is the sum of the exponents of the variables involved. For instance, in \(-5x\), the monomial is of degree 1 because \(x\) is raised to the first power. Thus, multiplying it with other terms often changes the degree of the resulting term while keeping the multiplication straightforward.

Combining Like Terms

Combining like terms is a process used in algebra to simplify expressions or equations. It involves adding or subtracting terms that have the same variable raised to the same power. This is essential when you want to tidy up your expression into its most simplified form.

In the final step of our given exercise, after applying the distributive property and multiplying, we have:

• -5x^3 + 15x^2 - 50x.

Notice that each term is unique in terms of the power of \(x\), so there are no like terms to combine. However, if there were multiple terms of the same degree (for instance, if there were two \(x^2\) terms), you would add or subtract the coefficients of these terms to condense them into a single term.

By combining like terms, you ensure your polynomial is as clear and concise as possible, making it easier to interpret or solve further.