Solving Polynomial Division | Long Division and Synthetic Division

solve.(x^3+3x^2-6x+2)-:(x-1)

To solve the expression (x^3+3x^2-6x+2)/ (x-1), we can use long division or synthetic division

To solve the expression (x^3+3x^2-6x+2)/ (x-1), we can use long division or synthetic division.

Using long division:

x^2 + 4x – 2
—————————-
x – 1 | x^3 + 3x^2 – 6x + 2

Step 1: Divide x into x^3 to get x^2 and write x^2 above the line.

x^2
—————————-
x – 1 | x^3 + 3x^2 – 6x + 2

Step 2: Multiply (x – 1) by x^2 to get x^3 – x^2 and subtract it from x^3 + 3x^2.

x^2 + 4x – 2
—————————-
x – 1 | x^3 + 3x^2 – 6x + 2
– (x^3 – x^2)
————–
4x^2 – 6x + 2

Step 3: Divide x into 4x^2 to get 4x and write 4x above the line.

x^2 + 4x – 2
—————————-
x – 1 | x^3 + 3x^2 – 6x + 2
– (x^3 – x^2)
————–
4x^2 – 6x + 2
– (4x^2 – 4x)
—————
-2x + 2

Step 4: Divide x into -2x to get -2 and write -2 above the line.

x^2 + 4x – 2
—————————-
x – 1 | x^3 + 3x^2 – 6x + 2
– (x^3 – x^2)
————–
4x^2 – 6x + 2
– (4x^2 – 4x)
—————
-2x + 2
– (-2x + 2)
————-
0

The result of the division is x^2 + 4x – 2 with no remainder. Therefore, the solution to the original expression (x^3 + 3x^2 – 6x + 2) / (x – 1) is x^2 + 4x – 2.

Definition of terms used:
– Long division: A method used to divide one polynomial by another polynomial.
– Synthetic division: A shortcut method for dividing polynomials, particularly when dividing by a binomial of the form (x – a).
– Polynomial: An expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication, but not division by a variable. For example, x^3 + 3x^2 – 6x + 2 is a polynomial.
– Dividend: The expression being divided. In this case, the dividend is (x^3 + 3x^2 – 6x + 2).
– Divisor: The expression used to divide the dividend. In this case, the divisor is (x – 1).
– Quotient: The result of the division. In this case, the quotient is x^2 + 4x – 2.
– Remainder: The leftover term after division, if any. In this case, there is no remainder.

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