What is the coefficient of x3  in   x(x2-1x)- x2(-1x-4)2?

To find the coefficient of \( x^3 \) in the expression:

\[
x(x^2 - 1x) - x^2(-1x - 4)^2
\]

Let's break it down step by step.

### Step 1: Simplify the first term
The first part of the expression is \( x(x^2 - 1x) \):

\[
x(x^2 - 1x) = x^3 - x^2
\]

### Step 2: Simplify the second term
The second part is \( x^2(-1x - 4)^2 \). First, expand \( (-1x - 4)^2 \):

\[
(-1x - 4)^2 = (x^2 + 8x + 16)
\]

Now, multiply this by \( x^2 \):

\[
x^2(x^2 + 8x + 16) = x^4 + 8x^3 + 16x^2
\]

### Step 3: Combine both parts
Now, subtract the second expression from the first one:

\[
(x^3 - x^2) - (x^4 + 8x^3 + 16x^2)
\]

Simplify by combining like terms:

\[
-x^4 + x^3 - 8x^3 - x^2 - 16x^2 = -x^4 - 7x^3 - 17x^2
\]

### Step 4: Identify the coefficient of \( x^3 \)
From the final expression:

\[
-x^4 - 7x^3 - 17x^2
\]

The coefficient of \( x^3 \) is **-7**.

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Would you like more details or further clarification on any step?

Here are 5 related questions:
1. How do you expand a binomial squared like \( (-1x - 4)^2 \)?
2. What are like terms, and how do you combine them in polynomial expressions?
3. Can you explain how multiplying polynomials works?
4. What is the degree of a polynomial and how is it determined?
5. How can you find the leading term of a polynomial?

**Tip:** When simplifying expressions, always follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

What is the coefficient of x^3 in x(x^2 - 1x) - x^2(-1x - 4)^2?

Learn how to find the coefficient of x^3 in the polynomial expression x(x^2 - 1x) - x^2(-1x - 4)^2 with a step-by-step breakdown. This problem involves polynomial expansion, combining like terms, and simplifying expressions. Ideal for students in Grades 9-12.

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