Polynomials Problem Help: How to Solve P(z)Q(z)=0 Using Coefficient Equations

In summary, the conversation discusses the steps needed to prove the converse of the statement that if P(z)Q(z)=0, then either P(z) or Q(z) is equal to 0. The conversation includes the use of equations, definitions, and logical reasoning to arrive at the conclusion that if P(z)Q(z) is not equal to 0, then P(z) and Q(z) are also not equal to 0.
  • #1
Ted123
446
0

Homework Statement



[PLAIN]http://img443.imageshack.us/img443/3096/questiond.jpg

The Attempt at a Solution



If [itex]P(z)Q(z)=0[/itex] then

[itex]\displaystyle a_0b_0 + (a_0b_1 + a_1 b_0)z + ... + \left( \sum_{i=0}^k a_i b_{k-i} \right) z^k + ... + a_n b_m z^{n+m} =0[/itex]

Now what? Equate coefficients?

[itex]a_0 b_0 =0 \Rightarrow a_0 = 0 \; \text{or}\; b_0 = 0[/itex]

[itex]a_0 = 0 \Rightarrow a_1 b_0 =0 \Rightarrow a_1 = 0 \; \text{or}\; b_0 =0[/itex]

[itex]b_0 =0 \Rightarrow a_0 b_1 = 0 \Rightarrow b_1 = 0 \; \text{or}\; a_0 = 0[/itex]

...
 
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  • #2
Hi Ted123! :smile:

Hint: start at zn+m. :wink:
 
  • #3


tiny-tim said:
Hi Ted123! :smile:

Hint: start at zn+m. :wink:

Doing this gives

[itex]a_n = 0\;\text{or}\; b_n = 0[/itex] but what next?

Would it be better to use the fact that:

If P isn't identically zero, it has at most n roots.
If Q isn't identically zero, it has at most m roots.

So, if neither P and Q are identically zero, PQ has at most m+n roots. How could I use this?
 
  • #4
Ted123 said:
… but what next?

think! :smile:

(how are an and bm defined?)
 
  • #5


tiny-tim said:
think! :smile:

(how are an and bm defined?)

Well in the question before this it states that [itex]a_n \neq 0[/itex] and [itex]b_m \neq 0[/itex] and it says in this question let P and Q be complex polynomials as in the previous question (I missed these conditions out when I copied the polynomials from the previous question).

So if [itex]a_n b_m =0[/itex] but [itex]a_n , b_m \neq 0[/itex] what do we have?
 
  • #6
a contradiction!

soooo … ? :smile:
 
  • #7


tiny-tim said:
a contradiction!

soooo … ? :smile:

[itex]P(z)Q(z) \neq 0[/itex]
 
  • #8
Yes … you've proved that if P and Q are not zero, then PQ is not zero. :smile:
 
  • #9


tiny-tim said:
Yes … you've proved that if P and Q are not zero, then PQ is not zero. :smile:

So does this prove the converse (which I need to prove) that if PQ=0 then P or Q are 0? Oh yes - I see it does now :smile:
 
Last edited:

What are polynomials?

Polynomials are mathematical expressions that consist of variables and coefficients, combined using the operations of addition, subtraction, and multiplication. They can have one or more terms, and the degree of a polynomial is determined by the highest exponent of the variable.

How do you solve polynomial problems?

Solving polynomial problems involves manipulating the expression using the properties of polynomials and solving for the variable or variables. This can be done by factoring, using the quadratic formula, or by synthetic division.

What is the difference between a monomial and a polynomial?

A monomial is a single term with a coefficient and a variable, while a polynomial has two or more terms. Monomials can be considered a special case of polynomials, where the degree is always one.

What are some real-world applications of polynomials?

Polynomials have many real-world applications, including in physics, engineering, and economics. They are used to model and solve various problems, such as calculating trajectories, determining optimal solutions, and predicting trends.

What are some important properties of polynomials?

Some important properties of polynomials include the degree, leading coefficient, and the number and type of roots. Additionally, polynomials have specific rules for addition, subtraction, and multiplication, and they follow the distributive property.

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