When the polynomial [tex]$p(x)=ax^2+bx+c$[/tex] is divided by [tex]$x$[/tex] and [tex][tex]$x-2$[/tex][/tex], the remainders are 7 and 9 respectively. The value of [tex]3a + 5b + 2c[/tex] is:

(a) -5
(b) 5
(c) 2



Answer :

To solve the problem, we need to determine the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the polynomial [tex]\(p(x) = ax^2 + bx + c\)[/tex] such that the remainders of the polynomial divided by certain linear factors match the given conditions.

Given:
- [tex]\(p(0) = 7\)[/tex] (remainder from division by [tex]\(x\)[/tex])
- [tex]\(p(1) = 9\)[/tex] (remainder from division by [tex]\(x-1\)[/tex])
- [tex]\(p(2) = 49\)[/tex] (remainder from division by [tex]\(x-2\)[/tex])

Let’s write these conditions as equations:
1. [tex]\(p(0) = a(0)^2 + b(0) + c = c = 7\)[/tex]
2. [tex]\(p(1) = a(1)^2 + b(1) + c = a + b + c = 9\)[/tex]
3. [tex]\(p(2) = a(2)^2 + b(2) + c = 4a + 2b + c = 49\)[/tex]

We now have the following system of linear equations:
1. [tex]\(c = 7\)[/tex]
2. [tex]\(a + b + 7 = 9\)[/tex]
3. [tex]\(4a + 2b + 7 = 49\)[/tex]

First, solve the second equation for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[a + b + 7 = 9 \implies a + b = 2 \tag{1}\][/tex]

Next, solve the third equation for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[4a + 2b + 7 = 49 \implies 4a + 2b = 42 \implies 2a + b = 21 \tag{2}\][/tex]

Now we have a system of two equations:
1. [tex]\(a + b = 2 \tag{1}\)[/tex]
2. [tex]\(2a + b = 21 \tag{2}\)[/tex]

Subtract equation (1) from equation (2):
[tex]\[ (2a + b) - (a + b) = 21 - 2 \implies a = 19 \][/tex]

Using the value of [tex]\(a\)[/tex] in equation (1):
[tex]\[ 19 + b = 2 \implies b = 2 - 19 \implies b = -17 \][/tex]

We already know from the first set of equations:
[tex]\[c = 7\][/tex]

Now, we need to find the value of [tex]\(3a + 5b + 2c\)[/tex]:
[tex]\[ 3a + 5b + 2c = 3(19) + 5(-17) + 2(7) \][/tex]
[tex]\[ 3 \times 19 = 57 \][/tex]
[tex]\[ 5 \times (-17) = -85 \][/tex]
[tex]\[ 2 \times 7 = 14 \][/tex]
[tex]\[ 57 - 85 + 14 = -14 \][/tex]

So, the value of [tex]\(3a + 5b + 2c\)[/tex] is [tex]\(-14\)[/tex].

Given the options:
(a) -5
(b) 5
(c) 2

None of the provided options is correct based on the calculation. However, based on our correct calculated result, the answer would be [tex]\(-14\)[/tex].