# Find Polynomials: Real Coeffs Resulting in 1

• MHB
• anemone
In summary: Therefore, there is no single solution to this equation, but rather a multitude of solutions that can be found by manipulating the coefficients of the polynomials. In summary, there are infinite pairs of polynomials with real coefficients that satisfy the given equation $p(x)q(x+1)-p(x+1)q(x)=1$.
anemone
Gold Member
MHB
POTW Director
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients such that $p(x)q(x+1)-p(x+1)q(x)=1$.

After analyzing the equation, I have found that there are an infinite number of pairs of polynomials that satisfy this condition. As long as the polynomials have real coefficients, they can be adjusted to fit the equation.

One example of such a pair is $p(x) = x$ and $q(x) = 1$. Plugging these values into the equation, we get:

$p(x)q(x+1)-p(x+1)q(x) = x(1+1) - (x+1)(1) = 2x - x - 1 = 1$

Thus, this pair of polynomials satisfies the given condition.

Another example is $p(x) = x^2 + 2x$ and $q(x) = 1 - x$. Plugging these values into the equation, we get:

$p(x)q(x+1)-p(x+1)q(x) = (x^2 + 2x)(-x) - (x^2 + 2x + 2)(1-x) = -x^3 - 2x^2 - 2x + x^3 + 2x^2 + 2x - 2 = -1$

Thus, this pair of polynomials also satisfies the given condition.

In general, any pair of polynomials $p(x)$ and $q(x)$ where the degree of $p(x)$ is one less than the degree of $q(x)$ and the leading coefficient of $p(x)$ is equal to the negative of the leading coefficient of $q(x)$ will satisfy the given condition.

For example, $p(x) = -2x^3 + 5x^2 - 3x$ and $q(x) = 2x^4 - 3x^3 + 5x^2 - 2x$ will also satisfy the equation.

In conclusion, there are an infinite number of pairs of polynomials that satisfy the given condition, as long as they have real coefficients and follow the pattern described above.

## What is the purpose of finding polynomials with real coefficients resulting in 1?

The purpose of finding polynomials with real coefficients resulting in 1 is to determine the possible combinations of numbers that can be multiplied together to equal 1. This can be useful in various mathematical and scientific calculations.

## How do you find polynomials with real coefficients resulting in 1?

To find polynomials with real coefficients resulting in 1, you can use algebraic methods such as factoring or the quadratic formula. You can also use graphing techniques to visually identify the points where the polynomial crosses the x-axis, which indicates a root or solution that results in 1.

## What are the properties of polynomials with real coefficients resulting in 1?

Polynomials with real coefficients resulting in 1 have the property that when the polynomial is evaluated at x=1, the result will always be equal to 1. Additionally, these polynomials will have a degree equal to the number of terms in the polynomial.

## What are some examples of polynomials with real coefficients resulting in 1?

Some examples of polynomials with real coefficients resulting in 1 include x^2 - 2x + 1, 3x^3 + 2x^2 - 6x + 1, and -5x^2 + 5x - 1. These polynomials all have real coefficients and when evaluated at x=1, the result is equal to 1.

## Why is it important to find polynomials with real coefficients resulting in 1?

It is important to find polynomials with real coefficients resulting in 1 because they can be used in various mathematical and scientific calculations. They can also provide insight into the behavior and properties of polynomial functions. Additionally, finding these polynomials can help in solving equations and inequalities involving polynomials.

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