Answer :
Let's find the sum of the given polynomials step-by-step.
We are adding two polynomials:
1. [tex]\( (5x^5 - 2x^3 + x) \)[/tex]
2. [tex]\( (4x^5 - 3x^4 + 2x^3 - 5x) \)[/tex]
To combine these polynomials, we'll add the coefficients of the like terms:
- For [tex]\(x^5\)[/tex]:
- Coefficients are [tex]\(5\)[/tex] and [tex]\(4\)[/tex].
- Sum: [tex]\(5 + 4 = 9x^5\)[/tex].
- For [tex]\(x^4\)[/tex]:
- The first polynomial has no [tex]\(x^4\)[/tex] term, so its coefficient is [tex]\(0\)[/tex].
- Coefficient of [tex]\(x^4\)[/tex] in the second polynomial is [tex]\(-3\)[/tex].
- Sum: [tex]\(0 - 3 = -3x^4\)[/tex].
- For [tex]\(x^3\)[/tex]:
- Coefficients are [tex]\(-2\)[/tex] and [tex]\(2\)[/tex].
- Sum: [tex]\(-2 + 2 = 0x^3\)[/tex]. This term cancels out since the sum is 0.
- For [tex]\(x^2\)[/tex]:
- Neither polynomial has an [tex]\(x^2\)[/tex] term, so the sum is [tex]\(0x^2\)[/tex]. No need to include this in the final result.
- For [tex]\(x\)[/tex]:
- Coefficients are [tex]\(1\)[/tex] and [tex]\(-5\)[/tex].
- Sum: [tex]\(1 - 5 = -4x\)[/tex].
- Constant Term:
- Neither polynomial has a constant term, so their sum is [tex]\(0\)[/tex].
Putting it all together, the sum of the polynomials is:
[tex]\[ 9x^5 - 3x^4 - 4x \][/tex]
We are adding two polynomials:
1. [tex]\( (5x^5 - 2x^3 + x) \)[/tex]
2. [tex]\( (4x^5 - 3x^4 + 2x^3 - 5x) \)[/tex]
To combine these polynomials, we'll add the coefficients of the like terms:
- For [tex]\(x^5\)[/tex]:
- Coefficients are [tex]\(5\)[/tex] and [tex]\(4\)[/tex].
- Sum: [tex]\(5 + 4 = 9x^5\)[/tex].
- For [tex]\(x^4\)[/tex]:
- The first polynomial has no [tex]\(x^4\)[/tex] term, so its coefficient is [tex]\(0\)[/tex].
- Coefficient of [tex]\(x^4\)[/tex] in the second polynomial is [tex]\(-3\)[/tex].
- Sum: [tex]\(0 - 3 = -3x^4\)[/tex].
- For [tex]\(x^3\)[/tex]:
- Coefficients are [tex]\(-2\)[/tex] and [tex]\(2\)[/tex].
- Sum: [tex]\(-2 + 2 = 0x^3\)[/tex]. This term cancels out since the sum is 0.
- For [tex]\(x^2\)[/tex]:
- Neither polynomial has an [tex]\(x^2\)[/tex] term, so the sum is [tex]\(0x^2\)[/tex]. No need to include this in the final result.
- For [tex]\(x\)[/tex]:
- Coefficients are [tex]\(1\)[/tex] and [tex]\(-5\)[/tex].
- Sum: [tex]\(1 - 5 = -4x\)[/tex].
- Constant Term:
- Neither polynomial has a constant term, so their sum is [tex]\(0\)[/tex].
Putting it all together, the sum of the polynomials is:
[tex]\[ 9x^5 - 3x^4 - 4x \][/tex]
