What is the degree of the polynomial shown in the table?

Question

A) Constant

B) Linear

C) Quadratic

D) Cubic

E) Quartic

D) Quintic

Answer 1

If you try linear

f(x)= ax+b

f(0) = 1 = b

f(1) = a+1=6 --> a= 5

f(2) = 5(2) +1 = 11 ≠ 29 No

If you try quadratic

f(x) = ax²+bx+1

f(1) = a+ b + 1 = 6 --> a+b = 5

f(2) = 4a +2b +1 = 23 --> 4a+2b = 22 --> a=2, b = 7

f(x) = 2x²+7b+1

f(3) = 18+21+1 =40 ≠ 88 No

If you try cubic

f(x) = ax³+bx²+cx+1

f(1) = a+b+c +1 = 6 --> a+b+c =5

f(2) = 8a+4b+2c+1 = 29 --> 8a+4b+2c = 28

f(3) = 81a+9b+3c+1 = 88 --> 81a+9b+3c = 87

Solving: a=3, b=0, c = 2

f(x)=3x³+2x+1

f(4)=192+8+1 = 201 Yes

f(5)=375+10+1=386 Yes

Answer D) Cubic

Answer 2

Since we are only asked the degree of the polynomial, do successive differencing until you get a constant. This is easily done with spreadsheets and allows for easy verification that it works everywhere. In the case of this problem, 3 differences are required, so it is a cubic.

Answer 3

Plotting this in a graphing calculator shows the right side values looks like a quadratic equation of y = ax², so the answer is a quadratic.

To prove this enter the values into the STAT Edit and create L1 and L2 values. Then perform the Stat Calc functionality and you will get:

ax² + bx + c with a = 54.96, b a = -309.49 and c = 529.62