Question

Free Response Question
A quartic function, h(x)=ax^4+x^3+19x^2-36, is represented numerically in the table of values above. h(x) has three distinct zeros, one of which has a multiplicity of two. Use the equation and the table to answer the equations that follow.

Answer 1

Table
x----h(x)
-3 -54
-2.5 -11
-2 0
-1.5 -6.75
1 -18
2 16
2.5 20.25
3 0
3.5 -60.5

Answer 2

Questions
a. Based on the values in the table, is a>0 or is a<0? Give a reason for your answer.
b. Use synthetic division and one of the zeros of h(x) from the table, find the value of a. Show your work.
c. Two of the zeros of h(x) are specifically listed in the table. Between what two values in the table does another zero h(x) exist? Give a numerical reason for your answer.
d. Based on the values in the table, which zero has a multiplicity of two? Give a numerical reason for your answer.

Answer 3

@bank2689
Do you know what a leading coefficient is?

Answer 4

Yes. I do. The number in front of a variable

Answer 5

Almost, it is the coefficient of the highest degree term of a single variable polynomial.
Can you identify the leading coefficient of the given quartic?

Answer 6

a

Answer 7

The number in front of a variable is a coefficient.
f(x)=4x^2+3x+2
All of 4, 3 and 2 are coefficients.
The leading coefficient of f(x) is 4 because x^2 is the highest degree term.

Answer 8

exactly.
Are you familiar with shapes of quartics according to the sign of "a"?

Answer 9

With all polynomials of even degree (0,2,4,6,...)
the shapes are like the following:

Do you notice the behaviours of +/- infinity when the sign of a changes?