Factoring equation with real coefficients

In summary, factoring equations with real coefficients involves breaking down an algebraic expression into smaller parts that can be multiplied together to get the original expression. This process is useful for solving equations and finding the roots of a polynomial. Real coefficients refer to numbers that are not imaginary or complex, such as whole numbers, fractions, and decimals. Factoring can also help with simplifying complicated expressions and identifying patterns within them. It is an important concept in algebra and can be used in various mathematical applications.
  • #1
Nathew

Homework Statement


Find the roots of [tex] z^4+4=0 [/tex] and use that to factor the expression into quadratic factors with real coefficients.

Homework Equations


DeMoivre's formula.

The Attempt at a Solution


I have been able to identify they are [tex] \pm 1 \pm i [/tex] but i have no idea how to factor the expression.
Thanks!
 
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  • #2
Nathew said:

Homework Statement


Find the roots of [tex] z^4+4=0 [/tex] and use that to factor the expression into quadratic factors with real coefficients.

Homework Equations


DeMoivre's formula.

The Attempt at a Solution


I have been able to identify they are [tex] \pm 1 \pm i [/tex] but i have no idea how to factor the expression.
Thanks!

No, they aren't. None of those are roots. Try them. Use deMoivre!
 
  • #3
Dick said:
No, they aren't. None of those are roots. Try them. Use deMoivre!

[tex] (1+i)^4+4=0 [/tex]
[tex] (-1+i)^4+4=0 [/tex]
[tex] (1-i)^4+4=0 [/tex]
[tex] (-1-i)^4+4=0 [/tex]

Not sure what you're talking about.
 
  • #4
Nathew said:

Homework Statement


Find the roots of [tex] z^4+4=0 [/tex] and use that to factor the expression into quadratic factors with real coefficients.
It is possible to do this in the reverse order. That is:
1. Factor ##\ z^4+4\ ## into quadratic factors with real coefficients.
then
2. find the roots of ##\ z^4+4=0\ .\ ##​

That's not following the instructions, but it may give some insight.

Suppose ##\ z^4+4=(z^2+az+2)(z^2+bz+2)\ ##.
Expand the right hand side & equate coefficients.
 
  • #5
Nathew said:
[tex] (1+i)^4+4=0 [/tex]
[tex] (-1+i)^4+4=0 [/tex]
[tex] (1-i)^4+4=0 [/tex]
[tex] (-1-i)^4+4=0 [/tex]

Not sure what you're talking about.

Sorry! I read your post as saying the roots were ##\pm 1## and ##\pm i##. If ##r_1## and ##r_2## are roots then ##(z-r_1)(z-r_2)## is a factor of your polynomial. Try multiplying that out when ##r_1## and ##r_2## are complex conjugates.
 
  • #6
Nathew said:

Homework Statement


Find the roots of [tex] z^4+4=0 [/tex] and use that to factor the expression into quadratic factors with real coefficients.

Homework Equations


DeMoivre's formula.

The Attempt at a Solution


I have been able to identify they are [tex] \pm 1 \pm i [/tex] but i have no idea how to factor the expression.
Thanks!
Yes, 1+ i, 1- i, -1- i, and -1+ i are roots so the we can write [itex]z^4+ 4= (z- (1+ i))(z- (1- i))(z- (-1+ i)(z- (-1- i))= (z- 1- i)(z- 1+ i)(z+ 1- i)(z+ 1+ i)[/itex]
Write [itex](z- 1- i)(z- 1+ i)= ((z- 1)- i)((z- 1)+ i)[/itex] and [itex](z+ 1- i)(z+ 1+ i)= ((z+ 1)- i)((z+ 1)+ i)[/itex]. Now use the fact that [itex](a- b)(a+ b)= a^2- b^2[/itex].
 
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  • #7
HallsofIvy said:
Yes, 1+ i, 1- i, -1- i, and -1+ i are roots so the we can write [itex]z^4+ 4= (z- (1+ i))(z- (1- i))(z- (-1+ i)(z- (-1- i))= (z- 1- i)(z- 1+ i)(z+ 1- i)(z+ 1+ i)[/itex]
Write [itex](z- 1- i)(z- 1+ i)= ((z- 1)- i)((z- 1)+ i)[/itex] and [itex](z+ 1- i)(z+ 1+ i)= ((z+ 1)- i)((z+ 1)+ i)[/itex]. Now use the fact that [itex](a- b)(a+ b)= a^2- b^2[/itex].
Thanks a lot this worked great!
 
  • #8
Nathew said:
Thanks a lot this worked great!
What did you get for the quadratic factors?
 
  • #9
SammyS said:
What did you get for the quadratic factors?
[tex](z^2-2z+2)(z^2+2z+2)[/tex]
 
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1. What is a real coefficient in a factoring equation?

A real coefficient in a factoring equation is a number that is multiplied by a variable. It is called a "real" coefficient because it is a rational number, meaning it can be expressed as a fraction and is not an imaginary number.

2. How do I know if an equation has real coefficients?

An equation has real coefficients if all the numbers in the equation are rational numbers, and there are no imaginary numbers (numbers with the square root of a negative number) present. You can also check by looking for the "i" symbol, which represents imaginary numbers.

3. Why is it important to factor an equation with real coefficients?

Factoring an equation with real coefficients can help us solve the equation and find its roots or solutions. It also helps us simplify complex equations and make them easier to understand and work with.

4. What are the steps to factor an equation with real coefficients?

The steps to factor an equation with real coefficients are: 1) Identify the highest common factor (HCF) of all the terms in the equation. 2) Use the distributive property to factor out the HCF. 3) Check if the remaining terms can be factored further. 4) Use the zero product property to find the solutions of the equation. 5) Verify the solutions by substituting them back into the original equation.

5. Can an equation with real coefficients have imaginary solutions?

No, an equation with real coefficients can only have real solutions. If an equation has imaginary solutions, it means that it has at least one imaginary coefficient, which contradicts the definition of a real coefficient. However, the process of factoring an equation with real coefficients may require us to use imaginary numbers in the intermediate steps.

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