Question

According to The Fundamental Theorem of Algebra, how many zeros does the function f(x) = 15x^17 + 41x^12 + 13x^3 – 10 have ?
Answer:
A)3
B)12
C)32
D)17
Please help meeee:)

Answer 1

What is the statement of the Fund Theorem of Algebra?

Answer 2

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed.

Sometimes, this theorem is stated as: every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors.

Answer 3

That's a fancy way of saying a first-order polynomial has 1 root
a second-order polynomial has 2 roots
a third-order polynomial has 3 roots
etc.

Given all that then, what order is your polynomial and how many roots must it have?

Answer 4

3

Answer 5

f(x) = 15x^17 + 41x^12 + 13x^3 – 10

The order of a polynomial in one variable x is the highest power of x in the polynomial.

For instance
x - 2
is first order, because the highest power of x is x^1 = x

x^2 - x - 2
is second order, because the highest power of x is x^2

15x^3 + 15x - 27
is third order, because the highest power of x is x^3

Hence given your polynomial
15x^17 + 41x^12 + 13x^3 – 10
what is its order?

Answer 6

i have no clue

Answer 7

What's the higher power of x in
f(x) = 15x^17 + 41x^12 + 13x^3 – 10

There are three choices:
x^17
x^12
x^3

Answer 8

17

Answer 9

Right. The highest power of x is x^17 ...

Answer 10

hence f(x) is a 17th-order polynomial ...

Answer 11

hence by the Fundamental Theorem of Algebra
f(x) = 15x^17 + 41x^12 + 13x^3 – 10

has 17 roots

Answer 12

Thank you<3