Which sequence has constant 2nd differences? (1 point)
A) {0.5, 5, 12.5, 23}
B) {5, 11.5, 18.5, 24.5}
C) {−0.5, 5, 13.5, 23}
D) {5, 13.5, 23, 34.5}
We can compute the second differences for each sequence to determine which one has constant second differences.
For sequence A:
First Differences: 5 - 0.5 = 4.5, 12.5 - 5 = 7.5, 23 - 12.5 = 10.5
Second Differences: 7.5 - 4.5 = 3, 10.5 - 7.5 = 3
The second differences are constant, so sequence A has constant second differences.
For sequence B:
First Differences: 11.5 - 5 = 6.5, 18.5 - 11.5 = 7, 24.5 - 18.5 = 6
Second Differences: 7 - 6.5 = 0.5, 6 - 7 = -1
The second differences are not constant, so sequence B does not have constant second differences.
For sequence C:
First Differences: 5 - (-0.5) = 5.5, 13.5 - 5 = 8.5, 23 - 13.5 = 9.5
Second Differences: 8.5 - 5.5 = 3, 9.5 - 8.5 = 1
The second differences are not constant, so sequence C does not have constant second differences.
For sequence D:
First Differences: 13.5 - 5 = 8.5, 23 - 13.5 = 9.5, 34.5 - 23 = 11.5
Second Differences: 9.5 - 8.5 = 1, 11.5 - 9.5 = 2
The second differences are not constant, so sequence D does not have constant second differences.
Therefore, the sequence with constant second differences is A) {0.5, 5, 12.5, 23}.
Using long division, determine which of the following polynomials divides evenly for a divisor of x+3.
A) x^3-5x^2+10x-15
B) 5x^2+7x-12
C) x^3-3x^2-13x+15
D) 3x^2-6x+9
To determine which polynomial divides evenly for a divisor of x+3, we can use long division.
Let's use the polynomial from option A, x^3-5x^2+10x-15, and divide it by x+3.
x^2 - 8x + 34
___________________
x + 3 | x^3 - 5x^2 + 10x - 15
- x^3 + 3x^2
_____________
- 8x^2 + 10x - 15
+ 8x^2 - 24x
_____________
- 14x - 15
+ 14x + 42
____________
27
The remainder is 27, which means x+3 does not evenly divide into x^3-5x^2+10x-15.
Next, let's use the polynomial from option B, 5x^2+7x-12, and divide it by x+3.
5x - 8
_______________
x + 3 | 5x^2 + 7x - 12
- 5x^2 + 15x
_______________
22x - 12
- 22x + 66
_______________
- 78
The remainder is -78, which means x+3 does not evenly divide into 5x^2+7x-12.
Next, let's use the polynomial from option C, x^3-3x^2-13x+15, and divide it by x+3.
x^2 - 4x + 1
___________________
x + 3 | x^3 - 3x^2 - 13x + 15
- x^3 + 3x^2
_____________
- 6x^2 - 13x + 15
+ 6x^2 - 18x
_____________
- 31x + 15
+ 31x + 93
____________
108
The remainder is 108, which means x+3 does not evenly divide into x^3-3x^2-13x+15.
Finally, let's use the polynomial from option D, 3x^2-6x+9, and divide it by x+3.
3
_____________
x + 3 | 3x^2 - 6x + 9
- 3x^2 + 9x
_____________
3x + 9
- 3x + 9
___________
18
The remainder is 18, which means x+3 does not evenly divide into 3x^2-6x+9.
Therefore, none of the given polynomials divide evenly for a divisor of x+3.