Question

Determine all polynomials \(P(x)\) such that $$ P\left(x^{2}+1\right)=(P(x))^{2}+1 \text { and } P(0)=0 . $$

Short answer

The polynomial \(P(x)\) that satisfies the given conditions is \(P(x) = x^2\).

Step-by-step solution

Substitute \(x=0\) in the equation

When substituted \(x = 0\) into the given equation, we get \(P(1) = P(0)^2 + 1=0^2+1=1\) since we know that \(P(0)=0.\)

Substitute \(x=1\) in the equation

Substituting \(x = 1\) into the original equation, we get \(P(2)=P(1)^2+1=1^2+1=2\) as we know \(P(1) = 1\) from step 1.

Determine the degree of polynomial

By examining \(P(x^{2} + 1) = (P(x))^2 +1\), we can deduce that the degree of \(P(x^{2} + 1)\) is even, meaning that the degree of \(P(x)\) is also even. Considering that the degree must be at least 2 (since we have \(P(1)\) and \(P(2)\) non-zero), the lowest degree we can have is 2. Therefore, we can consider proposing the polynomial \(P(x) = ax^2\).

Substitute the proposed polynomial into the equation

Substituting our proposed polynomial \(P(x) = ax^2\) into the original equation results in the following new equation : \(a(x^2+1)^2 = (ax^2)^2+1\). This simplifies to \(a(x^4 + 2x^2 + 1) = a^2x^4 + 1\). We can observe from this equation that \(a\) must equal to 1 for it to hold true. Thus, we can say \(P(x) = x^2\).

Verify that P(x) = x^2 holds for the original equation

Substituting \(P(x) = x^2\) into the original equation, we find that the equation remains valid. Therefore, we can conclude that \(P(x) = x^2\) is the solution to the given problem.

Key concepts

Polynomial Equations

A polynomial equation involves variables, coefficients, and exponents arranged in a specific way to create a polynomial expression set equal to a value, often zero. The expression is composed of terms that are either constants or the product of a constant and a variable raised to a non-negative integer power. These equations play a critical role in mathematics and form the basis of many algebraic problems.

For instance, the problem presented requires finding a polynomial function, let's call it \(P(x)\), that fits a specific condition: \( P\left(x^{2}+1\right)=(P(x))^{2}+1 \) and also satisfies \(P(0)=0\). This prompts students to analyze and manipulate the equation to solve for the unknown coefficients that define \(P(x)\). To successfully solve polynomial equations, one must apply various algebraic techniques, such as substitution, factoring, and sometimes employing the fundamental theorem of algebra, which asserts that a polynomial of degree \(n\) has exactly \(n\) roots or solutions when counted with multiplicity.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression when the expression is expressed in its standard form (i.e., terms are ordered from highest to lowest degree). It gives us a rough idea of the polynomial's behavior, especially for large values of the variable.

In the context of our exercise, determining the degree of \(P(x)\) is essential. The original equation indicates that the polynomial must have an even degree, as \(P(x^{2}+1)\) needs to match the even power in \((P(x))^2 + 1\). Knowing that the polynomial satisfies the conditions at \(P(1)\) and \(P(2)\), we infer that the degree cannot be zero, so the smallest possible even degree is 2. This critical piece of information narrows down our search for the correct polynomial considerably.

Functional Equations

Functional equations are equations where the unknowns are functions rather than simple quantities. These equations establish the relationship between the values of a function at different points. In our exercise, the functional equation is \(P\left(x^{2}+1\right)=(P(x))^{2}+1\), which requires us to find a function, or more specifically, a polynomial that satisfies this relationship for all inputs \(x\).

Handling functional equations often requires clever substitutions and a good understanding of function properties. For example, the step-by-step solution to the exercise demonstrates substituting specific values for \(x\) to narrow down the conditions that any potential solutions must meet. The process of verifying that \(P(x) = x^2\) is a solution to the functional equation further exemplifies the importance of understanding how both sides of the equation relate for all values of the variable.