Sure, let's break down each part of the problem step-by-step:
### 15a) Sum of the Polynomials
We are given the polynomials:
[tex]\[3x^2 - 2x + 4\][/tex]
and
[tex]\[-x^2 + 5x - 3\][/tex]
To find the sum of these polynomials, we simply add the corresponding coefficients of like terms (terms with the same power of [tex]\(x\)[/tex]).
1. Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[3x^2 + (-x^2) = 3x^2 - x^2 = 2x^2\][/tex]
2. Combine the [tex]\(x\)[/tex] terms:
[tex]\[-2x + 5x = 3x\][/tex]
3. Combine the constant terms:
[tex]\[4 + (-3) = 4 - 3 = 1\][/tex]
Putting these results together, the sum of the two polynomials is:
[tex]\[2x^2 + 3x + 1\][/tex]
### 15b) Difference of the Polynomials
Next, we are asked to find the difference between the polynomials [tex]\(3x^2 - 2x + 4\)[/tex] and [tex]\(-x^2 + 5x - 3\)[/tex].
To find the difference, we subtract the corresponding coefficients of the second polynomial from the first polynomial.
1. Subtract the [tex]\(x^2\)[/tex] terms:
[tex]\[3x^2 - (-x^2) = 3x^2 + x^2 = 4x^2\][/tex]
2. Subtract the [tex]\(x\)[/tex] terms:
[tex]\[-2x - 5x = -2x - 5x = -7x\][/tex]
3. Subtract the constant terms:
[tex]\[4 - (-3) = 4 + 3 = 7\][/tex]
Putting these results together, the difference of the two polynomials is:
[tex]\[4x^2 - 7x + 7\][/tex]
### Summary of Results
15a) The sum of the polynomials [tex]\(3x^2 - 2x + 4\)[/tex] and [tex]\(-x^2 + 5x - 3\)[/tex] is:
[tex]\[2x^2 + 3x + 1\][/tex]
15b) The difference of the polynomials [tex]\(3x^2 - 2x + 4\)[/tex] and [tex]\(-x^2 + 5x - 3\)[/tex] is:
[tex]\[4x^2 - 7x + 7\][/tex]
