Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ -2 x^{2}\left(4 x^{3}+5\right) $$

Short answer

-8x^5 - 10x^2

Step-by-step solution

Distribute the Constant

Distribute \(-2x^2\) to each term inside the parentheses. This will involve performing multiplication for each term:

Multiply \(x^2\) with Each Term

First, multiply \(x^2\) by \(4x^3\). This results in \[x^2 \times 4x^3 = 4x^{(2+3)} = 4x^5\]. Then, multiply \(x^2\) by \(5\), which is \[x^2 \times 5 = 5x^2\].

Multiply by the Coefficient \(-2\)

Now, multiply each of the results from Step 2 by \(-2\). For \[4x^5\], we have \[-2 \times 4x^5 = -8x^5\]. For \[5x^2\], we have \[-2 \times 5x^2 = -10x^2\].

Combine Like Terms

Since there are no like terms to combine, write the polynomial as the sum of the terms found in Step 3: \[-8x^5 - 10x^2\].

Key concepts

Polynomial Multiplication

Polynomial multiplication is an essential concept in algebra. It combines the terms of two polynomials to form a new polynomial. In the given exercise, we have \( -2 x^{2}(4 x^{3}+5) \).

To multiply polynomials, each term in the first polynomial is multiplied by each term in the second polynomial.

Here, we multiplied \( -2x^2 \) by each term inside the parentheses: \( 4x^3 \) and \( 5 \).

The process looked like this: \(-2x^2 \times 4x^3\), which equals \(-8x^5\).

Then, \(-2x^2 \times 5\), which equals \(-10x^2\).

So, the new polynomial after multiplication is \(-8x^5 - 10x^2\).

Distributive Property

The distributive property is a key tool for simplifying expressions and solving equations. It states that \( a(b + c) = ab + ac \).

In this exercise, the distributive property is used to multiply \(-2x^2\) with each term inside the parentheses \(4x^3 + 5\).

Here's how it works in detail:

• Distribute \(-2x^2\) to \(4x^3\) resulting in \(-2x^2 \times 4x^3 = -8x^5\).
  
  
• Next, distribute \(-2x^2\) to \(5\) resulting in \(-2x^2 \times 5 = -10x^2\).
  
  
  By using the distributive property, we can effectively break down more complex expressions into simpler parts.

Standard Form Polynomial

Polynomials are often written in standard form for clarity and ease of understanding. A polynomial in standard form arranges terms in decreasing order of the exponents.

For our completed polynomial, \(-8x^5 - 10x^2\), we see the terms are arranged from the highest power of \( x \) to the lowest.

Why is standard form important?

• It makes polynomials easier to read and compare.
  
  
• It helps in performing additional operations like addition, subtraction, and further multiplication.
  
  
• It is necessary for understanding polynomial functions and their graphs.

When dealing with polynomials, always aim to express your answer in standard form to ensure clarity and proper communication of your results.