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Answer :
Sure! Let's go through the process of synthetic division step-by-step to find the remainder.
Synthetic division is a method used to divide polynomials more quickly, especially when the divisor is of the form [tex]\(x - c\)[/tex].
In this problem, we are dividing the polynomial:
[tex]\[ f(x) = x^2 - 3x + 2 \][/tex]
by [tex]\( x - 1 \)[/tex]. Here, [tex]\( c = 1 \)[/tex].
Steps for Synthetic Division:
1. Set up the coefficients: Write down the coefficients of the polynomial in order, which are 1, -3, and 2.
2. Bring down the leading coefficient: Start by bringing down the first coefficient (1) as is.
3. Multiply and add: Multiply this number by [tex]\( c \)[/tex] (which is 1) and write the result under the next coefficient (-3). Then add this number to the second coefficient:
[tex]\[
1 \times 1 = 1 \quad (\text{Multiply})
\][/tex]
[tex]\[
-3 + 1 = -2 \quad (\text{Add})
\][/tex]
4. Continue the process: Repeat the multiplication and addition process with the result (-2).
[tex]\[
-2 \times 1 = -2 \quad (\text{Multiply})
\][/tex]
[tex]\[
2 + (-2) = 0 \quad (\text{Add})
\][/tex]
5. Read the result: The last number in the row of results is the remainder of the division.
So, after performing synthetic division, we find that the remainder is 0.
Therefore, the correct remainder for the division of the polynomial [tex]\( x^2 - 3x + 2 \)[/tex] by [tex]\( x - 1 \)[/tex] is 0.
Synthetic division is a method used to divide polynomials more quickly, especially when the divisor is of the form [tex]\(x - c\)[/tex].
In this problem, we are dividing the polynomial:
[tex]\[ f(x) = x^2 - 3x + 2 \][/tex]
by [tex]\( x - 1 \)[/tex]. Here, [tex]\( c = 1 \)[/tex].
Steps for Synthetic Division:
1. Set up the coefficients: Write down the coefficients of the polynomial in order, which are 1, -3, and 2.
2. Bring down the leading coefficient: Start by bringing down the first coefficient (1) as is.
3. Multiply and add: Multiply this number by [tex]\( c \)[/tex] (which is 1) and write the result under the next coefficient (-3). Then add this number to the second coefficient:
[tex]\[
1 \times 1 = 1 \quad (\text{Multiply})
\][/tex]
[tex]\[
-3 + 1 = -2 \quad (\text{Add})
\][/tex]
4. Continue the process: Repeat the multiplication and addition process with the result (-2).
[tex]\[
-2 \times 1 = -2 \quad (\text{Multiply})
\][/tex]
[tex]\[
2 + (-2) = 0 \quad (\text{Add})
\][/tex]
5. Read the result: The last number in the row of results is the remainder of the division.
So, after performing synthetic division, we find that the remainder is 0.
Therefore, the correct remainder for the division of the polynomial [tex]\( x^2 - 3x + 2 \)[/tex] by [tex]\( x - 1 \)[/tex] is 0.
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