Question

1. # If A Polynomial Function F(X) Has Roots 4 – 13I And 5, What Must Be A Factor Of F(X)?

Polynomials are an important part of mathematics, and understanding them can help you solve seemingly complex equations. In this article, we will explore the concept of a polynomial function and its roots, as well as discuss what must be a factor of the function in order to have these roots. By the end of this post, you should have a better understanding of polynomial functions and how they work. So let’s get started!

## What is a polynomial function?

A polynomial function is a function that can be written as the sum of a finite number of terms, each term consisting of a constant multiplied by a power of x. The powers must be non-negative integers. For example,

f(x) = 4x^3 + 3x^2 – 5x + 2

is a polynomial function. The roots of such a function are the values of x for which f(x) = 0. In this case, the roots are -1, 1/2, and 2.

## What are the roots of a polynomial function?

There are a few different ways to find the roots of a polynomial function, but one of the most common is to factor the function. This means that you take the function and break it down into factors that multiply together to equal the original function. In other words, if you have a polynomial function f(x) and you want to find its roots, you need to find what factors of f(x) will equal 0.

One way to do this is to use synthetic division. This is a method of dividing polynomials where you divide the original polynomial by one of its factors. If the resulting quotient has a root, then that factor must be a root of the original polynomial as well. So, for example, if you have a polynomial function f(x) = x^3 + 2x^2 + 5x + 6 and you want to find its roots, you could try synthetic division with each of its factors.

For x-6, we would get:

f(x)/(x-6) = x^2 + 8x + 36

Which does not have any real roots, so x-6 cannot be a factor of f(x). However, for x+2, we would get:

f(x)/(x+2) = x^2 – 4x + 12

Which does have a root (at x = –

## What is a factor of a polynomial function?

A factor of a polynomial function is a polynomial that when multiplied by another polynomial, results in the original polynomial. In other words, it is a number or variable that when multiplied by another produces the given polynomial. For example, x – 3 is a factor of x^2 – 3x + 9 because when x – 3 is multiplied by x + 3, the result is x^2 – 9.

## How to find the factors of a polynomial function

To find the factors of a polynomial function, you need to use the factoring formula. This formula states that if a polynomial function has roots i and j, then the factors of f(x) must be (x-i)(x-j). To use this formula, simply plug in the values for i and j into the equation. For example, if f(x) has roots 3 and 5, then the factors of f(x) would be (x-3)(x-5).

## Conclusion

In conclusion, if a polynomial function F(X) has roots 4 – 13I and 5, the factor of F(X) must be (x-4+13i)(x-5). Knowing this factor can help you find more accurate solutions to various problems. With this information in mind, you should now have a better understanding of how to solve related issues involving polynomial functions with complex roots.

2. Hmmm…if a polynomial function F(x) has roots 4 – 13i and 5, what must be a factor of F(x)?

Well, first off, let’s break it down a bit. A polynomial function is a mathematical expression that contains a sum of terms, each of which is the product of a constant number and a variable raised to a power. In this case, F(x) is the polynomial function.

The roots of a polynomial function are the values of x that make the function equal to zero. In this case, the roots of F(x) are 4 – 13i and 5, which means that when x = 4 – 13i or x = 5, F(x) = 0.

Now, the next step is to figure out what must be a factor of F(x). We can do this by factoring the polynomial into its individual terms. To do this, we need to write F(x) in its standard form.

F(x) = ax^2 + bx + c

In this case, the roots of F(x) are 4 – 13i and 5. So, we can use these values to substitute into the polynomial, and solve for a, b, and c.

4 – 13i:

F(4 – 13i) = 0

a(4 – 13i)^2 + b(4 – 13i) + c = 0
a(16 + 169i^2) + b(4 – 13i) + c = 0
16a + 169ai^2 + 4b – 13bi + c = 0

5:

F(5) = 0

a(5)^2 + b(5) + c = 0
25a + 5b + c = 0

We can then solve this system of equations to find the values of a, b, and c.

a = -1/20
b = -13/4
c = 6

This means that the factor of F(x) must be (x – 4 + 13i)(x – 5) = x^2 – 9x + 52i – 20.

So there you have it! If a polynomial function F(x) has roots 4 – 13i and 5, then the factor of F(x) must be (x – 4 + 13i)(x – 5) = x^2 – 9x + 52i – 20.