To find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for the given polynomials [tex]\(p(x)\)[/tex] and [tex]\(q(x)\)[/tex] such that [tex]\(p(x) = q(x)\)[/tex], we need to set the coefficients of corresponding powers of [tex]\(x\)[/tex] equal to each other.
Given:
[tex]\[
p(x) = (a + 2)x^3 + (b + 1)x^2 + (c + 3)x + 5
\][/tex]
[tex]\[
q(x) = (2a - 1)x^3 + (2b - 3)x^2 + (2c - 1)x
\][/tex]
Since [tex]\(p(x) = q(x)\)[/tex], the coefficients of [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term must be equal on both sides of the equation. This gives us the following system of equations:
1. Coefficient of [tex]\(x^3\)[/tex]:
[tex]\[
a + 2 = 2a - 1
\][/tex]
2. Coefficient of [tex]\(x^2\)[/tex]:
[tex]\[
b + 1 = 2b - 3
\][/tex]
3. Coefficient of [tex]\(x\)[/tex]:
[tex]\[
c + 3 = 2c - 1
\][/tex]
4. Constant term coefficients:
[tex]\[
5 = 0
\][/tex]
Let's solve these equations step-by-step.
### Step 1: Solve for [tex]\(a\)[/tex]
[tex]\[
a + 2 = 2a - 1
\][/tex]
Subtract [tex]\(a\)[/tex] from both sides:
[tex]\[
2 = a - 1
\][/tex]
Add 1 to both sides:
[tex]\[
3 = a
\][/tex]
### Step 2: Solve for [tex]\(b\)[/tex]
[tex]\[
b + 1 = 2b - 3
\][/tex]
Subtract [tex]\(b\)[/tex] from both sides:
[tex]\[
1 = b - 3
\][/tex]
Add 3 to both sides:
[tex]\[
4 = b
\][/tex]
### Step 3: Solve for [tex]\(c\)[/tex]
[tex]\[
c + 3 = 2c - 1
\][/tex]
Subtract [tex]\(c\)[/tex] from both sides:
[tex]\[
3 = c - 1
\][/tex]
Add 1 to both sides:
[tex]\[
4 = c
\][/tex]
### Step 4: Check the constant coefficients
The equation for the constant term:
[tex]\[
5 = 0
\][/tex]
This equation is contradictory and is not satisfied. Therefore, the given condition [tex]\(p(x) = q(x)\)[/tex] might impose that some non-zero constant term should be equal to zero, meaning one could infer either a possible misunderstanding, a special case, or an error in the problem formulation beyond the coefficients.
However, from the equations derived based on the polynomial coefficients, we can determine that the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] satisfying the other terms are:
[tex]\[
a = 3, \quad b = 4, \quad c = 4
\][/tex]
Thus, the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[
\boxed{a = 3, \, b = 4, \, c = 4}
\][/tex]
