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Answer :
We want to simplify the expression
[tex]$$
f(x) = \frac{3x^3 + 4x^2 - x + 2}{x-2}
$$[/tex]
by performing polynomial long division. Our goal is to express it in the form
[tex]$$
f(x) = Q(x) + \frac{R}{x-2},
$$[/tex]
where [tex]$Q(x)$[/tex] is the quotient polynomial and [tex]$R$[/tex] is the remainder.
Step 1. Divide the leading term
Divide the leading term of the numerator, [tex]$3x^3$[/tex], by the leading term of the denominator, [tex]$x$[/tex]:
[tex]$$
\frac{3x^3}{x} = 3x^2.
$$[/tex]
So, our first term in the quotient is [tex]$3x^2$[/tex].
Step 2. Multiply and subtract
Multiply the divisor by [tex]$3x^2$[/tex]:
[tex]$$
3x^2 \cdot (x-2) = 3x^3 - 6x^2.
$$[/tex]
Subtract this result from the original numerator:
[tex]$$
(3x^3 + 4x^2 - x + 2) - (3x^3 - 6x^2) = 10x^2 - x + 2.
$$[/tex]
Step 3. Repeat with the new numerator
Now, divide the new leading term [tex]$10x^2$[/tex] by [tex]$x$[/tex]:
[tex]$$
\frac{10x^2}{x} = 10x.
$$[/tex]
So, the next term in the quotient is [tex]$10x$[/tex].
Multiply the divisor by [tex]$10x$[/tex]:
[tex]$$
10x \cdot (x-2) = 10x^2 - 20x.
$$[/tex]
Subtract:
[tex]$$
(10x^2 - x + 2) - (10x^2 - 20x) = 19x + 2.
$$[/tex]
Step 4. Continue the division
Divide the new leading term [tex]$19x$[/tex] by [tex]$x$[/tex]:
[tex]$$
\frac{19x}{x} = 19.
$$[/tex]
Thus, the next term in the quotient is [tex]$19$[/tex].
Multiply the divisor by [tex]$19$[/tex]:
[tex]$$
19 \cdot (x-2) = 19x - 38.
$$[/tex]
Subtract:
[tex]$$
(19x + 2) - (19x - 38) = 40.
$$[/tex]
Since [tex]$40$[/tex] is a constant (degree [tex]$0$[/tex]) and the degree of the divisor [tex]$x-2$[/tex] is [tex]$1$[/tex], we cannot divide any further.
Step 5. Write the final expression
The quotient is
[tex]$$
Q(x) = 3x^2 + 10x + 19,
$$[/tex]
and the remainder is [tex]$R = 40$[/tex]. Therefore, the simplified form of [tex]$f(x)$[/tex] is:
[tex]$$
f(x) = 3x^2 + 10x + 19 + \frac{40}{x-2}.
$$[/tex]
This is the detailed step-by-step solution to the problem.
[tex]$$
f(x) = \frac{3x^3 + 4x^2 - x + 2}{x-2}
$$[/tex]
by performing polynomial long division. Our goal is to express it in the form
[tex]$$
f(x) = Q(x) + \frac{R}{x-2},
$$[/tex]
where [tex]$Q(x)$[/tex] is the quotient polynomial and [tex]$R$[/tex] is the remainder.
Step 1. Divide the leading term
Divide the leading term of the numerator, [tex]$3x^3$[/tex], by the leading term of the denominator, [tex]$x$[/tex]:
[tex]$$
\frac{3x^3}{x} = 3x^2.
$$[/tex]
So, our first term in the quotient is [tex]$3x^2$[/tex].
Step 2. Multiply and subtract
Multiply the divisor by [tex]$3x^2$[/tex]:
[tex]$$
3x^2 \cdot (x-2) = 3x^3 - 6x^2.
$$[/tex]
Subtract this result from the original numerator:
[tex]$$
(3x^3 + 4x^2 - x + 2) - (3x^3 - 6x^2) = 10x^2 - x + 2.
$$[/tex]
Step 3. Repeat with the new numerator
Now, divide the new leading term [tex]$10x^2$[/tex] by [tex]$x$[/tex]:
[tex]$$
\frac{10x^2}{x} = 10x.
$$[/tex]
So, the next term in the quotient is [tex]$10x$[/tex].
Multiply the divisor by [tex]$10x$[/tex]:
[tex]$$
10x \cdot (x-2) = 10x^2 - 20x.
$$[/tex]
Subtract:
[tex]$$
(10x^2 - x + 2) - (10x^2 - 20x) = 19x + 2.
$$[/tex]
Step 4. Continue the division
Divide the new leading term [tex]$19x$[/tex] by [tex]$x$[/tex]:
[tex]$$
\frac{19x}{x} = 19.
$$[/tex]
Thus, the next term in the quotient is [tex]$19$[/tex].
Multiply the divisor by [tex]$19$[/tex]:
[tex]$$
19 \cdot (x-2) = 19x - 38.
$$[/tex]
Subtract:
[tex]$$
(19x + 2) - (19x - 38) = 40.
$$[/tex]
Since [tex]$40$[/tex] is a constant (degree [tex]$0$[/tex]) and the degree of the divisor [tex]$x-2$[/tex] is [tex]$1$[/tex], we cannot divide any further.
Step 5. Write the final expression
The quotient is
[tex]$$
Q(x) = 3x^2 + 10x + 19,
$$[/tex]
and the remainder is [tex]$R = 40$[/tex]. Therefore, the simplified form of [tex]$f(x)$[/tex] is:
[tex]$$
f(x) = 3x^2 + 10x + 19 + \frac{40}{x-2}.
$$[/tex]
This is the detailed step-by-step solution to the problem.
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