Question

What is the simplified form of the quantity of 4 times t squared minus 5 times t plus three divided by the quantity of 2 times t squared minus t plus 1?

(1 point)
Responses solve it and tell me why

2 plus the quotient of the quantity negative 3 times t plus 1 over the quantity 2 times t squared minus t plus 1
Image with alt text: 2 plus the quotient of the quantity negative 3 times t plus 1 over the quantity 2 times t squared minus t plus 1

2 plus the quotient of the quantity negative 7 times t plus 5 over the quantity 2 times t squared minus t plus 1
Image with alt text: 2 plus the quotient of the quantity negative 7 times t plus 5 over the quantity 2 times t squared minus t plus 1

2 times t minus 5
Image with alt text: 2 times t minus 5

2 times t plus 3

Answer 1

[AI-generated answer — Bot GPT 3.5]

To simplify the given expression, we need to perform polynomial long division.

Given expression: (4t^2 - 5t + 3) / (2t^2 - t + 1)

Step 1: Divide the leading term of the numerator by the leading term of the denominator.

(4t^2 / 2t^2) = 2

Step 2: Multiply the entire denominator by the result obtained in Step 1 and subtract the result from the numerator.

(4t^2 - 5t + 3) - 2(2t^2 - t + 1) = -5t + 3 - 4t^2 + 2t - 2

Step 3: Repeat Steps 1 and 2 with the new polynomial.

(-5t + 3) / (2t^2 - t + 1)

(-5t / 2t) = -5/2

(-5t + 3) - (-5/2)(2t^2 - t + 1) = -5t + 3 + 5t^2/2 - 5t/2 + 5/2

Step 4: Repeat Steps 1 and 2 with the new polynomial.

(-5t + 3 + 5t^2/2 - 5t/2 + 5/2) / (2t^2 - t + 1)

This process will result in a complicated expression but essentially helps simplify the given quantity.

As a simplified result, we get: 2 + (-7t + 5) / (2t^2 - t + 1)