How Does Solving P(x) Influence Q(x^2)?

  • MHB
  • Thread starter anemone
  • Start date
In summary, the statement "Prove $P(x)=0 \implies Q(x^2)=0 | POTW #485 Oct 4, 2021" is a conditional statement where the truth of Q(x^2)=0 depends on the truth of P(x)=0. P(x)=0 represents a polynomial function with a root at x=0. The proof of this statement involves using mathematical principles and techniques and can be illustrated with an example. This statement is important in mathematics as it demonstrates the relationship between two statements and allows for the use of proof by contrapositive.
  • #1
anemone
Gold Member
MHB
POTW Director
3,883
115
Here is this week's POTW:

-----

Let $P(x)=x^3-2x+1$ and $Q(x)=x^3-4x^2+4x-1$. Show that if $P(r)=0$, then $Q(r^2)=0$.

-----

 
Physics news on Phys.org
  • #2
Congratulations to the following members for their correct solution!(Cool)

1. kaliprasad
2. lfdahl

Solution from lfdahl:
Given $P(r) = r^3-2r+1 = 0$:

\[0 = [P(r)]^2 = \left [ r^3-2r+1 \right ]^2 \\\\ = r^6-4r^4+2r^3+4r^2-4r+1 \\\\ = \underbrace{r^6 - 4r^4 + 4r^2-1}_{= Q(r^2)} +\underbrace{2r^3-4r + 2}_{= 2P(r)} = Q(r^2)\;\;\;\; q.e.d.\]
 

FAQ: How Does Solving P(x) Influence Q(x^2)?

What does the notation "P(x)=0" mean?

The notation "P(x)=0" means that the polynomial function P(x) is equal to zero. In other words, all the terms in the polynomial have been simplified and the resulting expression equals zero.

How does the statement "P(x)=0" relate to "Q(x^2)=0"?

The statement "P(x)=0" implies that all values of x that make P(x) equal to zero are also solutions for Q(x^2)=0. This is because when x is squared, the resulting value will still make P(x) equal to zero.

Can you give an example of a polynomial function that satisfies "P(x)=0" but not "Q(x^2)=0"?

Yes, for example, let P(x) = x^2 - 4x + 4. This polynomial satisfies P(x)=0 when x=2, but it does not satisfy Q(x^2)=0 when x=2, as Q(x^2) would result in a non-zero value.

How can you prove "P(x)=0 \implies Q(x^2)=0"?

To prove "P(x)=0 \implies Q(x^2)=0", you can use a direct proof by assuming that P(x)=0 and then showing that this implies Q(x^2)=0. This can be done by substituting x^2 for x in P(x) and simplifying the resulting expression to show that it equals zero.

What implications does this statement have in mathematics and science?

This statement has implications in various areas of mathematics and science, particularly in the fields of algebra, calculus, and physics. It can be used to solve equations and systems of equations, as well as to prove theorems and make mathematical predictions. In science, this statement can be applied to model and analyze real-world phenomena, such as in physics equations that involve polynomial functions and their solutions.

Similar threads

MHB Prove: No Real Roots for Polynomial x^4-px^3+qx^2-sqrt(q)x+1
Replies
2
Views
1K
MHB Sum of Cosine Values for $x$ | POTW #480 8/24/2021 | $100^{\circ}<x<200^{\circ}$
Replies
1
Views
1K
MHB Can you solve this week's POTW #474: Four real numbers with unique permutations?
Replies
1
Views
1K
MHB How to Tackle the Latest POTW Inequality Challenge?
Replies
2
Views
1K
MHB Can x(x+y) Be Proved to Be ≤ 2 for Positive Real Numbers?
Replies
11
Views
2K
MHB Can You Prove the Polynomial Has Three Distinct Roots?
Replies
1
Views
1K
MHB Find the Minimum Values of Monic Quadratic Polynomials - POTW #403 Feb 2nd, 2020
Replies
2
Views
2K
MHB Is Sine of x Greater Than or Equal to 2x Over Pi for Angles Up to Pi/2?
Replies
1
Views
1K
MHB Can You Solve This Quartic Equation with Roots Involving Square-Free Integers?
Replies
1
Views
1K
MHB How Do You Calculate the Sum of Minimum Values in Polynomials?
Replies
1
Views
1K

Recent Insights

Back
Top