High School

Simplify the expression:

[tex]\left(5x^4 - 4x^3 - 2x^2 + x - 19\right) - \left(x^4 + 5x^3 + 8x^2 + x + 5\right)[/tex]

Choose the correct simplified form:

A. [tex]4x^4 - 9x^3 - 10x^2 - 24[/tex]

B. [tex]4x^4 - 9x^3 - 10x^2 - 2x - 24[/tex]

C. [tex]4x^4 - 9x^3 - 10x^2 + 2x - 24[/tex]

D. [tex]-4x^4 - 9x^3 - 10x^2 - 24[/tex]

Answer :

To solve the expression [tex]\((5x^4 - 4x^3 - 2x^2 + x - 19) - (x^4 + 5x^3 + 8x^2 + x + 5)\)[/tex], we need to subtract the terms of the second polynomial from the corresponding terms of the first polynomial. Here are the steps to do it:

1. Identify the Terms of Each Polynomial:

- First polynomial: [tex]\(5x^4 - 4x^3 - 2x^2 + x - 19\)[/tex]
- Second polynomial: [tex]\(x^4 + 5x^3 + 8x^2 + x + 5\)[/tex]

2. Subtract the Corresponding Terms:

- Subtract the [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - x^4 = 4x^4\)[/tex]
- Subtract the [tex]\(x^3\)[/tex] terms: [tex]\(-4x^3 - 5x^3 = -9x^3\)[/tex]
- Subtract the [tex]\(x^2\)[/tex] terms: [tex]\(-2x^2 - 8x^2 = -10x^2\)[/tex]
- Subtract the [tex]\(x\)[/tex] terms: [tex]\(x - x = 0\)[/tex]
- Subtract the constant terms: [tex]\(-19 - 5 = -24\)[/tex]

3. Combine the Results:

After performing the subtraction for each term, we combine them to form the resulting polynomial:

[tex]\[
4x^4 - 9x^3 - 10x^2 + 0x - 24
\][/tex]

4. Simplify the Result:

Since the coefficient of [tex]\(x\)[/tex] is 0, we can write the result as:

[tex]\[
4x^4 - 9x^3 - 10x^2 - 24
\][/tex]

Therefore, the simplified polynomial after subtracting the second polynomial from the first is [tex]\(4x^4 - 9x^3 - 10x^2 - 24\)[/tex].