Answer :
Sure! Let's solve the problem by subtracting the polynomials using a vertical format, step by step.
We have two polynomials:
1. The polynomial from which we will subtract:
[tex]\(-x^5 + 2x^4 - 3x^3 + 4x^2 + 5x\)[/tex]
2. The polynomial to subtract:
[tex]\(5x^4 + 4x^3 - 3x^2 + 2x\)[/tex]
We will align them vertically based on the degree of each term:
```
-x^5 + 2x^4 - 3x^3 + 4x^2 + 5x
-( 5x^4 + 4x^3 - 3x^2 + 2x)
--------------------------------
```
Now, let's subtract the coefficients for each term:
- Degree 5: [tex]\(-x^5 - 0 \cdot x^5 = -x^5\)[/tex]
- Degree 4: [tex]\(2x^4 - 5x^4 = -3x^4\)[/tex]
- Degree 3: [tex]\(-3x^3 - 4x^3 = -7x^3\)[/tex]
- Degree 2: [tex]\(4x^2 - (-3x^2) = 4x^2 + 3x^2 = 7x^2\)[/tex]
- Degree 1: [tex]\(5x - 2x = 3x\)[/tex]
Putting these results together, the final result of the subtraction is:
[tex]\[
-x^5 - 3x^4 - 7x^3 + 7x^2 + 3x
\][/tex]
Looking at the options, the correct answer is:
(a) [tex]\(-x^5 - 3x^4 - 7x^3 + 7x^2 + 3x\)[/tex]
We have two polynomials:
1. The polynomial from which we will subtract:
[tex]\(-x^5 + 2x^4 - 3x^3 + 4x^2 + 5x\)[/tex]
2. The polynomial to subtract:
[tex]\(5x^4 + 4x^3 - 3x^2 + 2x\)[/tex]
We will align them vertically based on the degree of each term:
```
-x^5 + 2x^4 - 3x^3 + 4x^2 + 5x
-( 5x^4 + 4x^3 - 3x^2 + 2x)
--------------------------------
```
Now, let's subtract the coefficients for each term:
- Degree 5: [tex]\(-x^5 - 0 \cdot x^5 = -x^5\)[/tex]
- Degree 4: [tex]\(2x^4 - 5x^4 = -3x^4\)[/tex]
- Degree 3: [tex]\(-3x^3 - 4x^3 = -7x^3\)[/tex]
- Degree 2: [tex]\(4x^2 - (-3x^2) = 4x^2 + 3x^2 = 7x^2\)[/tex]
- Degree 1: [tex]\(5x - 2x = 3x\)[/tex]
Putting these results together, the final result of the subtraction is:
[tex]\[
-x^5 - 3x^4 - 7x^3 + 7x^2 + 3x
\][/tex]
Looking at the options, the correct answer is:
(a) [tex]\(-x^5 - 3x^4 - 7x^3 + 7x^2 + 3x\)[/tex]
