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Find a polynomial of the specified degree that has the given zero. Degree 4 with zeros -4, 3, 0, and -2.

This question aims to find the polynomial with a degree 4 and given zeros of -4, 3, 0, and -2.

The question depends on the concepts of polynomial expressions and the degree of polynomials with zeros. The degree of any polynomial is the highest exponent of its independent variable. The zeros of a polynomial are the values where the output of the polynomial becomes zero.

Expert Answer

If c is the zero of the polynomial, then (x-c) is a factor of the polynomial if and only if the polynomial is zero at c. Let the polynomial we need to find is P(x). Then -4, 3, 0, and -2 will be the zeros of P(x). We can conclude that:

\[ c = -4\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x + 4)\ is\ a\ factor\ of\ P(x) \]

\[ c = 3\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x\ -\ 3)\ is\ a\ factor\ of\ P(x) \]

\[ c = 0\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x\ -\ 0)\ is\ a\ factor\ of\ P(x) \]

\[ c = -2\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x + 2)\ is\ a\ factor\ of\ P(x) \]

We can write that polynomial P(x) is equal to the product of its factors according to the factor theorem. The expression for P(x) is given as:

\[ P(x) = ( x + 4 )( x\ -\ 3 )( x\ -\ 0 )( x + 2 ) \]

\[ P(x) = x( x + 2 )( x\ -\ 3 )( x + 4 ) \]

Simplifying the equation will give us the polynomial P(x).

\[ P(x) = (x^2 + 2x )( x^2 + x\ -\ 12) \]

\[ P(x) = x^4 + 3x^3\ -\ 10x^2\ -\ 24x \]

Numerical Result

The polynomial P(x) with degree 4 and zeros -4, 3, 0, and -2 is calculated to be:

\[ P(x) = x^4 + 3x^3\ -\ 10x^2\ -\ 24x \]

Example

Find a polynomial with degree 3 and zeros -1, 0, and 1.

Let P(x) is the polynomial function with a degree of 3. It has zeros of -1, 0, and 1. So following must be true for the polynomial P(x).

\[ c = -1\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x + 1)\ is\ a\ factor\ of\ P(x) \]

\[ c = 1\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x\ -\ 1)\ is\ a\ factor\ of\ P(x) \]

\[ c = 0\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x\ -\ 0)\ is\ a\ factor\ of\ P(x) \]

We can write the P(x) equal to its factors as:

\[ P(x) = x( x + 1 )( x\ -\ 1 ) \]

\[ P(x) = x( x^2\ -\ x + x\ -\ 1 ) \]

\[ P(x) = x( x^2\ -\ 1 ) \]

\[ P(x) = x^3\ -\ x \]

The polynomial P(x) has a degree of 3.

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