What is the constant term of the quotient when 2x^3-3x^2 + 4x-2 is divided by x^2-x+2?
1 answer:
Answer:
-1
Step-by-step explanation:
See the attachment for the polynomial long division. The constant in the quotient is -1.
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Here, there is a remainder of -x. If there were no remainder the constant in the quotient is the ratio of the constant in the dividend to the constant in the divisor: -2/2 = -1.
That could be a first guess in a "guess and check" solution approach.
<em>Guess</em>: first term of binomial quotient is (2x^3)/x^2 = 2x; last term of binomial quotient is -2/2 = -1. So, the quotient is guessed to be (2x -1).
<em>Check</em>: (2x -1)(x^2 -x +2) = 2x^3 -3x^2 +5x -2
Subtracting this from the actual dividend gives a remainder of -x. This has a lower degree than the divisor, so no further adjustment of the quotient is required.
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That's an easy parallelogram, parallel to the x axis. Let's label the points
A(-4,-1),B(-2,4),C(4,4),D(2,-1)
AD and BC are the sides parallel to the x axis. We'll call those the base, it has length 4 - -2 = 6.
The height is the change in y between the bases, so 4 - -1 = 5.
The area is base times height, 6 × 5 = 30
Answer: 30
The shoelace formula gives us the area of a polygon from 2D coordinates without worrying about the shape:
(-4,-1),(-2,4),(4,4),(2,-1)
(-2,4),(4,4),(2,-1),(-4,-1)