High School

What is the difference of the polynomials?

[tex]\left(5x^3 + 4x^2\right) - \left(6x^2 - 2x - 9\right)[/tex]

A. [tex]-x^3 + 6x^2 + 9[/tex]

B. [tex]-x^3 + 2x^2 - 9[/tex]

C. [tex]5x^3 - 2x^2 - 2x - 9[/tex]

D. [tex]5x^3 - 2x^2 + 2x + 9[/tex]

Answer :

Sure! Let's work through the problem of finding the difference between the polynomials step by step.

We're given two polynomials:

1. [tex]\( 5x^3 + 4x^2 \)[/tex]
2. [tex]\( 6x^2 - 2x - 9 \)[/tex]

The problem asks us to find the difference between these two polynomials, which means we need to subtract the second polynomial from the first.

Here's how to do it:

### Step 1: Set up the Subtraction

Write down the expression for the subtraction:

[tex]\( (5x^3 + 4x^2) - (6x^2 - 2x - 9) \)[/tex]

### Step 2: Distribute the Negative Sign

When subtracting a polynomial, you must distribute the negative sign across the entire second polynomial:

[tex]\( 5x^3 + 4x^2 - 6x^2 + 2x + 9 \)[/tex]

### Step 3: Combine Like Terms

Combine like terms by adding or subtracting the coefficients of terms with the same degree:

- The [tex]\(x^3\)[/tex] term: [tex]\(5x^3\)[/tex] remains unchanged since there's no other [tex]\(x^3\)[/tex] term to combine with.
- The [tex]\(x^2\)[/tex] term: [tex]\(4x^2 - 6x^2 = -2x^2\)[/tex]
- The [tex]\(x\)[/tex] term: [tex]\(0x + 2x = 2x\)[/tex] (there's no [tex]\(x\)[/tex] term in the first polynomial, hence it's treated as [tex]\(0x\)[/tex] initially)
- The constant term: [tex]\(+9\)[/tex] remains unchanged as there's no other constant to combine with.

### Step 4: Write the Result

After combining the terms, the resulting polynomial is:

[tex]\( 5x^3 - 2x^2 + 2x + 9 \)[/tex]

That's your final answer for the difference of the polynomials!