1. What polynomial has a graph that passes through the given points?
(-2, 2) (-1, -1) (1, 5) (3, 67)
A. y = -x3 + 4x2 + 2x - 2
B. y = x3 + 4x2 + 2x - 2
C. y = x4 + 4x3 + 2x2 - 2x
D. y = x3 - 4x2 - 2x - 2
B?
2. Write -2x 2(-5x 2 +4x 3) in standard form.
A.
-7x5 -10x4
B.
-8x5 -20x4
C.
-7x +2x4
D.
-8x5 +10x4
C ?
To find the correct answer for the first question, we need to determine which polynomial matches the given points.
First, let's substitute the x and y values into each of the options and see which one satisfies all the points.
For option A:
When x = -2, y = -(-2)^3 + 4(-2)^2 + 2(-2) - 2 = -8 + 16 - 4 - 2 = 2 (satisfies the first point)
When x = -1, y = -(-1)^3 + 4(-1)^2 + 2(-1) - 2 = -1 + 4 - 2 - 2 = -1 (satisfies the second point)
When x = 1, y = -(1)^3 + 4(1)^2 + 2(1) - 2 = -1 + 4 + 2 - 2 = 3 (does not satisfy the third point)
When x = 3, y = -(3)^3 + 4(3)^2 + 2(3) - 2 = -27 + 36 + 6 - 2 = 13 (does not satisfy the fourth point)
Option A does not satisfy all of the given points.
For option B:
When x = -2, y = (-2)^3 + 4(-2)^2 + 2(-2) - 2 = -8 + 16 - 4 - 2 = 2 (satisfies the first point)
When x = -1, y = (-1)^3 + 4(-1)^2 + 2(-1) - 2 = -1 + 4 - 2 - 2 = -1 (satisfies the second point)
When x = 1, y = (1)^3 + 4(1)^2 + 2(1) - 2 = 1 + 4 + 2 - 2 = 5 (satisfies the third point)
When x = 3, y = (3)^3 + 4(3)^2 + 2(3) - 2 = 27 + 36 + 6 - 2 = 67 (satisfies the fourth point)
Option B satisfies all of the given points. Therefore, the correct answer is option B: y = x^3 + 4x^2 + 2x - 2.
For the second question, we need to simplify the expression -2x^2(-5x^2 + 4x^3) into standard form.
Using the distributive property, we can multiply -2x^2 by -5x^2 and 4x^3:
-2x^2(-5x^2 + 4x^3) = -2x^2(-5x^2) + (-2x^2)(4x^3)
Simplifying further:
= 10x^4 - 8x^5
The expression -2x^2(-5x^2 + 4x^3) in standard form is -8x^5 + 10x^4. Hence, the correct answer is option D: -8x^5 + 10x^4.
Therefore, the correct answers are:
1. B. y = x^3 + 4x^2 + 2x - 2
2. D. -8x^5 + 10x^4