Find a polynominal function f(x) of least degrees having only the real coeficents and zeros as given

In summary, a polynomial with real coefficients will have complex zeroes occurring in conjugate pairs. So if two of the zeroes are $5+i$ and $4-i$, the other ones will be $5-i$ and $4+i$. Hence $f(x)$ will be a polynomial of degree at least 4.
  • #1
karush
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MHB
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Find a polynominal function f(x) of least degrees having only the real coeficents and zeros as given

$$5+i \quad 4-i$$
ok I did this but don't think this is the final answer
$(x-(5+i))(x-(5-i))=x^2+26$
$(x-(4-i))(x-(4+i))=x^2-17$
 
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  • #2
You've got the right idea, but your quadratic products aren't correct. :)
 
  • #3
MarkFL said:
You've got the right idea, but your quadratic products aren't correct. :)

$$(x-(5+i))(x-(5-i))=x^2+26$$
$$(x-(4-i))(x-(4+i))=x^2+17$$

is this as far as can go?

or is it

(
x^2+26)(x^2+17)
 
  • #4
You want to first get two quadratics as you've done, and then the quartic polynomial will be their product, but you don't have the correct quadratics yet. Check your multiplications...you should have \(x\) terms. :)
 
  • #5
MarkFL said:
You want to first get two quadratics as you've done, and then the quartic polynomial will be their product, but you don't have the correct quadratics yet. Check your multiplications...you should have \(x\) terms. :)
$$(x^2 - 10 x + 26)(x^2 - 8 x +17)=x^4-18x^3+123x^2-378x+442$$
 
  • #6
karush said:
Find a polynominal function f(x) of least degrees having only the real coeficents and zeros as given

$$5+i \quad 4-i$$
ok I did this but don't think this is the final answer
$(x-(5+i))(x-(5-i))=x^2+26$
$(x-(4-i))(x-(4+i))=x^2-17$

A polynomial with real coefficients will have complex zeroes occurring in conjugate pairs. So if two of the zeroes are $5+i$ and $4-i$, the other ones will be $5-i$ and $4+i$. Hence $f(x)$ will be a polynomial of degree at least 4.

A polynomial with zeroes $a\pm bi$ is $x^2-2ax+(a^2+b^2)$. Hence
$$f(x)\ =\ k(x^2-10x+26)(x^2-8x+17)$$
where $k$ is any nonzero real number.
 
  • #7
What are some good online calculators for hunting down zeros

I thot EMath was nice and you
Pull latex from it.
 

Related to Find a polynominal function f(x) of least degrees having only the real coeficents and zeros as given

1. What is a polynomial function?

A polynomial function is a mathematical function that is made up of terms consisting of constants and variables raised to non-negative integer powers. It is represented in the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where an to a0 are the coefficients and n is the degree of the polynomial.

2. What does it mean for a polynomial function to have only real coefficients?

A polynomial function is said to have only real coefficients if all the coefficients in its terms are real numbers, meaning they do not contain any imaginary numbers. This is important because it ensures that the function will only have real solutions or zeros.

3. Why is it important for a polynomial function to have only real zeros?

Having only real zeros means that the solutions to the function are real numbers, which can be easily understood and interpreted. Additionally, real zeros also ensure that the graph of the function will intersect the x-axis at those points, making it easier to visualize the behavior of the function.

4. How do you find the degree of a polynomial function?

The degree of a polynomial function is determined by the highest exponent on the variable in its terms. For example, if a polynomial function has terms with x2, x3, and x4, its degree would be 4. If a term with x does not exist, the degree is considered to be 0.

5. How do you find a polynomial function of least degree with given zeros?

To find a polynomial function of least degree with given zeros, you can use the fact that a polynomial with real coefficients will have conjugate complex roots. This means that if a + bi is a zero, then a - bi is also a zero. Knowing this, you can find the factors of the polynomial by using (x - a) and (x - b) for each zero, and then multiply them all together to get the final polynomial function.

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