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If, on dividing the polynomial [tex]\(f(x)=2x^4-3x^2-ax+b\)[/tex] by [tex]\((x-1)\)[/tex] and [tex]\((x+1)\)[/tex], the remainders are -3 and 7 respectively, then find the remainder when [tex]\(f(x)\)[/tex] is divided by [tex]\((x-2)\)[/tex].

Sagot :

GinnyAnswer
Certainly! Let's work through the problem step-by-step.

We are given the polynomial:

[tex]\[ f(x) = 2x^4 - 3x^2 - ax + b \][/tex]

and the remainders when it is divided by [tex]\( (x-1) \)[/tex] and [tex]\( (x+1) \)[/tex]:

- The remainder when divided by [tex]\( (x-1) \)[/tex] is -3, which means [tex]\( f(1) = -3 \)[/tex].
- The remainder when divided by [tex]\( (x+1) \)[/tex] is 7, which means [tex]\( f(-1) = 7 \)[/tex].

First, we set up the equations based on these conditions.

### Step 1:
Evaluate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 2(1)^4 - 3(1)^2 - a(1) + b \][/tex]
[tex]\[ f(1) = 2 - 3 - a + b \][/tex]
Given [tex]\( f(1) = -3 \)[/tex]:
[tex]\[ 2 - 3 - a + b = -3 \][/tex]
Simplifying this:
[tex]\[ -1 - a + b = -3 \][/tex]
[tex]\[ -a + b = -2 \quad \text{(Equation 1)}\][/tex]

### Step 2:
Evaluate [tex]\( f(-1) \)[/tex]:
[tex]\[ f(-1) = 2(-1)^4 - 3(-1)^2 - a(-1) + b \][/tex]
[tex]\[ f(-1) = 2 - 3 + a + b \][/tex]
Given [tex]\( f(-1) = 7 \)[/tex]:
[tex]\[ 2 - 3 + a + b = 7 \][/tex]
Simplifying this:
[tex]\[ -1 + a + b = 7 \][/tex]
[tex]\[ a + b = 8 \quad \text{(Equation 2)}\][/tex]

### Step 3:
Solve the system of linear equations (Equation 1) and (Equation 2):

From Equation 2:
[tex]\[ b = 8 - a \][/tex]

Substitute [tex]\( b \)[/tex] in Equation 1:
[tex]\[ -a + (8 - a) = -2 \][/tex]
[tex]\[ -a + 8 - a = -2 \][/tex]
[tex]\[ -2a + 8 = -2 \][/tex]
[tex]\[ -2a = -2 - 8 \][/tex]
[tex]\[ -2a = -10 \][/tex]
[tex]\[ a = 5 \][/tex]

Substitute [tex]\( a \)[/tex] in Equation 2:
[tex]\[ 5 + b = 8 \][/tex]
[tex]\[ b = 8 - 5 \][/tex]
[tex]\[ b = 3 \][/tex]

So, we have [tex]\( a = 5 \)[/tex] and [tex]\( b = 3 \)[/tex].

### Step 4:
Find the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( (x-2) \)[/tex].

Evaluate [tex]\( f(2) \)[/tex]:

[tex]\[ f(x) = 2x^4 - 3x^2 - 5x + 3 \][/tex]

Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 2(2)^4 - 3(2)^2 - 5(2) + 3 \][/tex]
[tex]\[ f(2) = 2(16) - 3(4) - 5(2) + 3 \][/tex]
[tex]\[ f(2) = 32 - 12 - 10 + 3 \][/tex]
[tex]\[ f(2) = 32 - 12 = 10 \][/tex]
[tex]\[ f(2) = 10 - 10 = 0 \][/tex]
[tex]\[ f(2) = 0 + 3 = 13 \][/tex]

Thus, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( (x-2) \)[/tex] is:

[tex]\[ \boxed{13} \][/tex]
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