Sure, let's solve the problem step-by-step.
Firstly, we have three polynomials that we want to add together:
[tex]\[ P_1(x) = x^3 - 5x^2 + 7 \][/tex]
[tex]\[ P_2(x) = 3x^2 + 5x - 2 \][/tex]
[tex]\[ P_3(x) = 2x^3 - x + 7 \][/tex]
Our goal is to find the resulting polynomial when these three are added together:
[tex]\[ P(x) = P_1(x) + P_2(x) + P_3(x) \][/tex]
Let's add these polynomials term by term.
1. Combine the [tex]\( x^3 \)[/tex] terms:
[tex]\[ x^3 + 2x^3 = 3x^3 \][/tex]
2. Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[ -5x^2 + 3x^2 = -2x^2 \][/tex]
3. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ 5x - x = 4x \][/tex]
4. Combine the constant terms:
[tex]\[ 7 - 2 + 7 = 12 \][/tex]
Now, let's write out the resulting polynomial by compiling these terms together:
[tex]\[ P(x) = 3x^3 - 2x^2 + 4x + 12 \][/tex]
Hence, the resulting polynomial when you add [tex]\( (x^3 - 5x^2 + 7) \)[/tex], [tex]\( (3x^2 + 5x - 2) \)[/tex], and [tex]\( (2x^3 - x + 7) \)[/tex] is:
[tex]\[ 3x^3 - 2x^2 + 4x + 12 \][/tex]
