Algebra 2 Urgent question: ABOUT WRITING POLYNOMIAL FUNCTIONS?

Question

Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1

1, -4, 5

Also how would this work because of the "i"?

-2, -2, 3, -4i

Answer 1

(x - 1)(x + 4)(x - 5) =
(x - 1)(x^2 - x - 20) =
x^3 - 3x^2 - 21x + 20 = 0

Answer 2

Polynomial function theorems for zeros
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Polynomial function theorems for zeros are a set of theorems aiming to find (or determine the nature) of the complex zeros of a polynomial function.

Found in most precalculus textbooks, these theorems include:

* Remainder theorem
* Factor theorem
* Descartes' rules of signs
* Rational zeros theorem
* Bounds on zeros theorem also known as the boundedness theorem
* Intermediate value theorem
* Conjugate pairs theorem

[edit] Background

A polynomial function is a function of the form

p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0 ,\,

where a_i\, (i = 0, 1, 2, ..., n) are complex numbers and a_n \ne 0.

If p(z) = anzn + an − 1zn − 1 + ... + a2z2 + a1z + a0 = 0, then z is called a zero of p(x). If z is real, then z is a real zero of p(x); if z is imaginary, the z is a complex zero of p(x), although complex zeros include both real and imaginary zeros.

The fundamental theorem of algebra states that every polynomial function of degree n \ge 1 has at least one complex zero. It follows that every polynomial function of degree n \ge 1 has exactly ncomplex zeros, not necessarily distinct.

* If the degree of the polynomial function is 1, i.e., p(x) = a_1 x + a_0 \,, then its (only) zero is \frac{-a_0}{a_1}.
* If the degree of the polynomial function is 2, i.e., p(x) = a_2 x^2 + a_1 x + a_0 \,, then its two zeros (not necessarily distinct) are \frac{-a_1 + \sqrt{{a_1}^2 - 4 a_2 a_0}}{2 a_2} and \frac{-a_1 - \sqrt{{a_1}^2 - 4 a_2 a_0}}{2 a_2}.

A degree one polynomial is also known as a linear function, whereas a degree two polynomial is also known as a quadratic function and its two zeros are merely a direct result of the quadratic formula. However, difficulty rises when the degree of the polynomial, n, is higher than 2. It is true that there is a cubic formula for a cubic function (a degree three polynomial) and there is a quartic formula for a quartic function (a degree four polynomial), but they are very complicate. To make matter worst, there is no general formula for a polynomial function of degree 5 or higher.

[edit] The theorems

[edit] Remainder theorem

The remainder theorem states that if p(x) is divided by x − c, then the remainder is p(c).
For example, when p(x) = x3 + 2x − 3 is divided by x − 2, the remainder (if we don't care about the quotient) will be p(2) = 23 + 2(2) − 3 = 9. When p(x) is divided by x + 1, the remainder is p( − 1) = ( − 1)3 + 2( − 1) − 3 = − 6. However, this theorem is most useful when the remainder is 0 since it will yield a zero of p(x). For example, p(x) is divided by x − 1, the remainder is p(1) = (1)3 + 2(1) − 3 = 0, so 1 is a zero of p(x) (by the definition of zero of a polynomial function).

Answer 3

Do the opposite of what you do when you factor and solve a polynomial. If you saw that an equation factored to (x-1)(x+4)(x-5)=0, you would conclude that there were zeros at 1, -4, and 5.

So to solve this problem, simply multiply out (x-1)(x+4)(x-5). Bingo, you have your function.