Answer :
To find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for the polynomial [tex]\(P(x) = 5(x-1)(x-2)(x-3) + a(x-1)(x-2) + b(x-1) + c\)[/tex], we use the given conditions for the remainders when [tex]\(P(x)\)[/tex] is evaluated at [tex]\(x = 1\)[/tex], [tex]\(x = 2\)[/tex], and [tex]\(x = 3\)[/tex].
### Step-by-step solution:
1. Evaluate [tex]\(P(x)\)[/tex] at [tex]\(x = 1\)[/tex]:
Given that [tex]\(P(1) = 7\)[/tex], substitute [tex]\(x = 1\)[/tex] into [tex]\(P(x)\)[/tex]:
[tex]\[ P(1) = 5(1-1)(1-2)(1-3) + a(1-1)(1-2) + b(1-1) + c \][/tex]
Simplify the expression:
[tex]\[ P(1) = 5(0)(-1)(-2) + a(0)(-1) + b(0) + c = 0 + 0 + 0 + c = c \][/tex]
Therefore, we have:
[tex]\[ c = 7 \][/tex]
2. Evaluate [tex]\(P(x)\)[/tex] at [tex]\(x = 2\)[/tex]:
Given that [tex]\(P(2) = 2\)[/tex], substitute [tex]\(x = 2\)[/tex] into [tex]\(P(x)\)[/tex]:
[tex]\[ P(2) = 5(2-1)(2-2)(2-3) + a(2-1)(2-2) + b(2-1) + c \][/tex]
Simplify the expression:
[tex]\[ P(2) = 5(1)(0)(-1) + a(1)(0) + b(1) + c = 0 + 0 + b + c = b + c \][/tex]
Substitute [tex]\(c = 7\)[/tex] from the previous step:
[tex]\[ b + 7 = 2 \implies b = 2 - 7 = -5 \][/tex]
3. Evaluate [tex]\(P(x)\)[/tex] at [tex]\(x = 3\)[/tex]:
Given that [tex]\(P(3) = 1\)[/tex], substitute [tex]\(x = 3\)[/tex] into [tex]\(P(x)\)[/tex]:
[tex]\[ P(3) = 5(3-1)(3-2)(3-3) + a(3-1)(3-2) + b(3-1) + c \][/tex]
Simplify the expression:
[tex]\[ P(3) = 5(2)(1)(0) + a(2)(1) + b(2) + c = 0 + 2a + 2b + c = 2a + 2b + c \][/tex]
Substitute [tex]\(b = -5\)[/tex] and [tex]\(c = 7\)[/tex] from the previous steps:
[tex]\[ 2a + 2(-5) + 7 = 1 \implies 2a - 10 + 7 = 1 \implies 2a - 3 = 1 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ 2a = 4 \implies a = 2 \][/tex]
### Conclusion:
Thus, the values of the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[ a = 2, \quad b = -5, \quad c = 7 \][/tex]
### Step-by-step solution:
1. Evaluate [tex]\(P(x)\)[/tex] at [tex]\(x = 1\)[/tex]:
Given that [tex]\(P(1) = 7\)[/tex], substitute [tex]\(x = 1\)[/tex] into [tex]\(P(x)\)[/tex]:
[tex]\[ P(1) = 5(1-1)(1-2)(1-3) + a(1-1)(1-2) + b(1-1) + c \][/tex]
Simplify the expression:
[tex]\[ P(1) = 5(0)(-1)(-2) + a(0)(-1) + b(0) + c = 0 + 0 + 0 + c = c \][/tex]
Therefore, we have:
[tex]\[ c = 7 \][/tex]
2. Evaluate [tex]\(P(x)\)[/tex] at [tex]\(x = 2\)[/tex]:
Given that [tex]\(P(2) = 2\)[/tex], substitute [tex]\(x = 2\)[/tex] into [tex]\(P(x)\)[/tex]:
[tex]\[ P(2) = 5(2-1)(2-2)(2-3) + a(2-1)(2-2) + b(2-1) + c \][/tex]
Simplify the expression:
[tex]\[ P(2) = 5(1)(0)(-1) + a(1)(0) + b(1) + c = 0 + 0 + b + c = b + c \][/tex]
Substitute [tex]\(c = 7\)[/tex] from the previous step:
[tex]\[ b + 7 = 2 \implies b = 2 - 7 = -5 \][/tex]
3. Evaluate [tex]\(P(x)\)[/tex] at [tex]\(x = 3\)[/tex]:
Given that [tex]\(P(3) = 1\)[/tex], substitute [tex]\(x = 3\)[/tex] into [tex]\(P(x)\)[/tex]:
[tex]\[ P(3) = 5(3-1)(3-2)(3-3) + a(3-1)(3-2) + b(3-1) + c \][/tex]
Simplify the expression:
[tex]\[ P(3) = 5(2)(1)(0) + a(2)(1) + b(2) + c = 0 + 2a + 2b + c = 2a + 2b + c \][/tex]
Substitute [tex]\(b = -5\)[/tex] and [tex]\(c = 7\)[/tex] from the previous steps:
[tex]\[ 2a + 2(-5) + 7 = 1 \implies 2a - 10 + 7 = 1 \implies 2a - 3 = 1 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ 2a = 4 \implies a = 2 \][/tex]
### Conclusion:
Thus, the values of the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[ a = 2, \quad b = -5, \quad c = 7 \][/tex]
