Answer :
Certainly! Let's carefully solve the problem step by step:
We're given an expression to simplify:
[tex]\[
(5x^4 - 4x^3 - 2x^2 + x - 19) - (x^4 + 5x^3 + 8x^2 + x + 5)
\][/tex]
To simplify this, we'll subtract the second polynomial from the first. This involves changing the signs of each of the terms in the second polynomial and then combining like terms. Here's how it works:
1. Rewrite the expression by distributing the negative sign through the second polynomial:
[tex]\[
5x^4 - 4x^3 - 2x^2 + x - 19 - x^4 - 5x^3 - 8x^2 - x - 5
\][/tex]
2. Combine like terms:
- [tex]$x^4$[/tex] terms: [tex]\(5x^4 - x^4 = 4x^4\)[/tex]
- [tex]$x^3$[/tex] terms: [tex]\(-4x^3 - 5x^3 = -9x^3\)[/tex]
- [tex]$x^2$[/tex] terms: [tex]\(-2x^2 - 8x^2 = -10x^2\)[/tex]
- [tex]$x$[/tex] terms: [tex]\(x - x = 0\)[/tex]
- Constant terms: [tex]\(-19 - 5 = -24\)[/tex]
3. Write the simplified expression:
[tex]\[
4x^4 - 9x^3 - 10x^2 - 24
\][/tex]
That's the fully simplified expression! Each term has been combined correctly, following polynomial arithmetic rules.
We're given an expression to simplify:
[tex]\[
(5x^4 - 4x^3 - 2x^2 + x - 19) - (x^4 + 5x^3 + 8x^2 + x + 5)
\][/tex]
To simplify this, we'll subtract the second polynomial from the first. This involves changing the signs of each of the terms in the second polynomial and then combining like terms. Here's how it works:
1. Rewrite the expression by distributing the negative sign through the second polynomial:
[tex]\[
5x^4 - 4x^3 - 2x^2 + x - 19 - x^4 - 5x^3 - 8x^2 - x - 5
\][/tex]
2. Combine like terms:
- [tex]$x^4$[/tex] terms: [tex]\(5x^4 - x^4 = 4x^4\)[/tex]
- [tex]$x^3$[/tex] terms: [tex]\(-4x^3 - 5x^3 = -9x^3\)[/tex]
- [tex]$x^2$[/tex] terms: [tex]\(-2x^2 - 8x^2 = -10x^2\)[/tex]
- [tex]$x$[/tex] terms: [tex]\(x - x = 0\)[/tex]
- Constant terms: [tex]\(-19 - 5 = -24\)[/tex]
3. Write the simplified expression:
[tex]\[
4x^4 - 9x^3 - 10x^2 - 24
\][/tex]
That's the fully simplified expression! Each term has been combined correctly, following polynomial arithmetic rules.
