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Answer :
To find the remainder when [tex]\( f(x) = 4x^3 - 12x^2 + 14x - 3 \)[/tex] is divided by [tex]\( g(x) = 2x - 1 \)[/tex], we can use the Remainder Theorem.
The Remainder Theorem states that if a polynomial [tex]\( f(x) \)[/tex] is divided by a linear divisor of the form [tex]\( x - c \)[/tex], then the remainder of this division is equal to [tex]\( f(c) \)[/tex].
In this case, the divisor [tex]\( g(x) = 2x - 1 \)[/tex] can be rewritten in the form of [tex]\( x - c \)[/tex] by setting it equal to zero:
[tex]\[ 2x - 1 = 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ 2x = 1 \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
So, to find the remainder, we evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = \frac{1}{2} \)[/tex]:
[tex]\[ f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^3 - 12\left(\frac{1}{2}\right)^2 + 14\left(\frac{1}{2}\right) - 3 \][/tex]
Calculating each term:
1. [tex]\( 4\left(\frac{1}{2}\right)^3 = 4 \times \frac{1}{8} = \frac{1}{2} \)[/tex]
2. [tex]\(- 12\left(\frac{1}{2}\right)^2 = -12 \times \frac{1}{4} = -3\)[/tex]
3. [tex]\( 14\left(\frac{1}{2}\right) = 14 \times \frac{1}{2} = 7 \)[/tex]
4. [tex]\(- 3\)[/tex]
Adding these up:
[tex]\[\frac{1}{2} - 3 + 7 - 3 = \frac{1}{2} + 4 - 3 = \frac{1}{2} + 1 = \frac{3}{2} \][/tex]
Thus, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( g(x) = 2x - 1 \)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
The Remainder Theorem states that if a polynomial [tex]\( f(x) \)[/tex] is divided by a linear divisor of the form [tex]\( x - c \)[/tex], then the remainder of this division is equal to [tex]\( f(c) \)[/tex].
In this case, the divisor [tex]\( g(x) = 2x - 1 \)[/tex] can be rewritten in the form of [tex]\( x - c \)[/tex] by setting it equal to zero:
[tex]\[ 2x - 1 = 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ 2x = 1 \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
So, to find the remainder, we evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = \frac{1}{2} \)[/tex]:
[tex]\[ f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^3 - 12\left(\frac{1}{2}\right)^2 + 14\left(\frac{1}{2}\right) - 3 \][/tex]
Calculating each term:
1. [tex]\( 4\left(\frac{1}{2}\right)^3 = 4 \times \frac{1}{8} = \frac{1}{2} \)[/tex]
2. [tex]\(- 12\left(\frac{1}{2}\right)^2 = -12 \times \frac{1}{4} = -3\)[/tex]
3. [tex]\( 14\left(\frac{1}{2}\right) = 14 \times \frac{1}{2} = 7 \)[/tex]
4. [tex]\(- 3\)[/tex]
Adding these up:
[tex]\[\frac{1}{2} - 3 + 7 - 3 = \frac{1}{2} + 4 - 3 = \frac{1}{2} + 1 = \frac{3}{2} \][/tex]
Thus, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( g(x) = 2x - 1 \)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
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