Question
What is the simpler form of the following expression?
(6x3−x2+1)÷(2x+1)
show your work
(6x3−x2+1)÷(2x+1)
show your work
Answers
GPT 3.5
To simplify the expression (6x^3 - x^2 + 1) ÷ (2x + 1), we need to use long division.
First, divide the first term of the numerator by the first term of the denominator:
6x^3 ÷ 2x = 3x^2
Multiply the entire denominator by 3x^2:
3x^2 * (2x + 1) = 6x^3 + 3x^2
Subtract this result from the original expression to get the new remainder:
(6x^3 - x^2 + 1) - (6x^3 + 3x^2) = -4x^2 + 1
Now, divide the first term of the new remainder by the first term of the denominator:
-4x^2 ÷ 2x = -2x
Multiply the entire denominator by -2x:
-2x * (2x + 1) = -4x^2 - 2x
Subtract this result from the previous remainder:
(-4x^2 + 1) - (-4x^2 - 2x) = 2x + 1
Since 2x + 1 is the remainder and it is less than 2x + 1, the final simplified form of the expression is:
3x^2 - 2x + (2x + 1) / (2x + 1) = 3x^2 - 2x + 1
Therefore, the simpler form of the expression is 3x^2 - 2x + 1.
First, divide the first term of the numerator by the first term of the denominator:
6x^3 ÷ 2x = 3x^2
Multiply the entire denominator by 3x^2:
3x^2 * (2x + 1) = 6x^3 + 3x^2
Subtract this result from the original expression to get the new remainder:
(6x^3 - x^2 + 1) - (6x^3 + 3x^2) = -4x^2 + 1
Now, divide the first term of the new remainder by the first term of the denominator:
-4x^2 ÷ 2x = -2x
Multiply the entire denominator by -2x:
-2x * (2x + 1) = -4x^2 - 2x
Subtract this result from the previous remainder:
(-4x^2 + 1) - (-4x^2 - 2x) = 2x + 1
Since 2x + 1 is the remainder and it is less than 2x + 1, the final simplified form of the expression is:
3x^2 - 2x + (2x + 1) / (2x + 1) = 3x^2 - 2x + 1
Therefore, the simpler form of the expression is 3x^2 - 2x + 1.