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].
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].
