Using The Fundamental Theorem Of Algebra, Determine The Total Number Of Roots Of The Polynomial Function Using Its Factored Form. F ( X ) = ( X + 1 ) ( X − 3 ) ( X − 4 F(x) = (x+1)(x-3)(x-4 F ( X ) = ( X + 1 ) ( X − 3 ) ( X − 4 ]What Is The Total Number Of Roots?

Mar 10, 2025 by ADMIN 264 views

**Understanding the Fundamental Theorem of Algebra and Its Application to Polynomial Functions**

What is the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra is a fundamental concept in algebra that states that every non-constant polynomial equation of degree n has exactly n complex roots. This theorem is a cornerstone in the field of algebra and has numerous applications in various areas of mathematics, including calculus, geometry, and number theory.

What is a Polynomial Function?

A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial function are typically represented by x, and the coefficients are constants. A polynomial function can be written in the form:

f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, and a_0 are coefficients, and n is the degree of the polynomial.

What is the Factored Form of a Polynomial Function?

The factored form of a polynomial function is an expression in which the polynomial is written as a product of linear factors. Each linear factor is of the form (x - r), where r is a root of the polynomial. The factored form of a polynomial function can be written as:

f(x) = (x - r_1)(x - r_2)...(x - r_n)

where r_1, r_2, ..., r_n are the roots of the polynomial.

How to Determine the Total Number of Roots of a Polynomial Function Using Its Factored Form?

To determine the total number of roots of a polynomial function using its factored form, we need to count the number of linear factors. Each linear factor corresponds to a root of the polynomial. Therefore, the total number of roots of a polynomial function is equal to the number of linear factors in its factored form.

Example: Determining the Total Number of Roots of a Polynomial Function

Consider the polynomial function:

f(x) = (x+1)(x-3)(x-4)

To determine the total number of roots of this polynomial function, we need to count the number of linear factors. In this case, there are three linear factors: (x+1), (x-3), and (x-4). Therefore, the total number of roots of this polynomial function is 3.

Q&A

Q: What is the Fundamental Theorem of Algebra? A: The Fundamental Theorem of Algebra is a fundamental concept in algebra that states that every non-constant polynomial equation of degree n has exactly n complex roots.

Q: What is a polynomial function? A: A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What is the factored form of a polynomial function? A: The factored form of a polynomial function is an expression in which the polynomial is written as a product of linear factors.

Q: How to determine the total number of roots of a polynomial function using its factored form? A: To determine the total number of roots of a polynomial function using its factored form, we need to count the number of linear factors. Each linear factor corresponds to a root of the polynomial.

Q: What is the total number of roots of the polynomial function f(x) = (x+1)(x-3)(x-4)? A: The total number of roots of the polynomial function f(x) = (x+1)(x-3)(x-4) is 3.

Conclusion

In conclusion, the Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n complex roots. To determine the total number of roots of a polynomial function using its factored form, we need to count the number of linear factors. Each linear factor corresponds to a root of the polynomial. By understanding the Fundamental Theorem of Algebra and its application to polynomial functions, we can determine the total number of roots of a polynomial function using its factored form.

References

[1] Fundamental Theorem of Algebra. (n.d.). In Encyclopedia Britannica.
[2] Polynomial Function. (n.d.). In Math Open Reference.
[3] Factored Form of a Polynomial Function. (n.d.). In Math Is Fun.