Question

Show that \((x-9)^{4}\) is not congruent to \(\left(x^{4}-9\right)\) modulo 4.

Short answer

It has been confirmed that \((x-9)^{4}\) is not congruent to \(\left(x^{4}-9\right)\) modulo 4 since their modulo 4 forms are \(x^4 + 2x^2 + 1(mod 4)\) and \(x^{4}-1(mod 4)\) respectively, and these are not equal.

Step-by-step solution

Expanding \((x-9)^{4}\)

Using the binomial theorem, expand \((x-9)^{4}\) to get \(x^4 - 4x^3*9 + 6x^2*81 -4x*729 + 6561\).

Applying modulo operation on \((x-9)^{4}\)

Take each term of the resultant expression modulo 4. Doing this operation, you get \(x^4 - 0 + 2x^2*1- 0 + 1 = x^4 + 2x^2 + 1 (mod 4)\).

Applying modulo operation on \(\left(x^{4}-9\right)\)

Take \(\left(x^{4}-9\right)\) modulo 4. You get: \(x^{4}-1\) (because 9 modulo 4 equals to 1).

Comparing the results

Compare the two results: \(x^4 + 2x^2 + 1(mod 4)\) and \(x^{4}-1(mod 4)\). You see that the two are not the same, so \((x-9)^{4}\) is not congruent to \(\left(x^{4}-9\right)\) modulo 4.

Key concepts

Understanding the Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form \((a + b)^n\). It states that:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• Every term in the expansion is a product of a binomial coefficient, a power of \(a\), and a power of \(b\).

This formula is essential when dealing with polynomials raised to a power. To expand \((x-9)^4\), we use:

• \(a = x\), \(b = -9\), and \(n = 4\).
• Apply the coefficients from Pascal’s triangle or use the binomial coefficient formula \(\binom{4}{k}\).

The expanded form will include terms like \(x^4\), \(-4x^3*9\), and so on. This step-by-step expansion process helps us prepare the expression for modular arithmetic.

Exploring Congruence Relations

Congruence relations deal with the idea of equivalence in modular arithmetic. When we say \(a \equiv b \pmod{m}\), it means that the difference \(a-b\) is divisible by \(m\). Here’s how it applies:

• Evaluate each term of a polynomial modulo the chosen number (in this case, 4).
• This simplifies calculations by reducing each term to a smaller equivalence class modulo 4.

For example, in step 2 of the solution, the expression \((x-9)^4\) is transformed by taking each term modulo 4, resulting in \(x^4 + 2x^2 + 1 \pmod{4}\). This is critical for comparing expressions in modular arithmetic and determining congruence.

The Essence of Polynomial Expansion

Polynomial expansion involves expressing a power of a binomial as a sum of terms. This is closely related to the binomial theorem.

• It converts complex polynomial forms into simpler ones.
• Each term in the expansion represents a distinct part of the polynomial makeup.

When working with \((x-9)^4\), the expansion yields several terms with various powers of \(x\) and constants. After expanding, applying modulus helps simplify further, making congruence comparison possible. This procedure is essential when showing differences between polynomial expressions under modular arithmetic, as seen in the given exercise. By comparing these expansions, we can determine if they are congruent under a specified modulus.