Question

The coefficients \(a_{l}\) in the polynomial \(Q(x)=a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x\) are all integers. Show that \(Q(n)\) is divisible by 24 for all integers \(n \geq 0\) if and only if all the following conditions are satisfied: (i) \(2 a_{4}+a_{3}\) is divisible by 4 ; (ii) \(a_{4}+a_{2}\) is divisible by 12 ; (iii) \(a_{4}+a_{3}+a_{2}+a_{1}\) is divisible by 24 .

Short answer

Q(x) is divisible by 24 for all n ≥ 0 if (i) 2a4 + a3 is divisible by 4, (ii) a4 + a2 is divisible by 12, (iii) a4 + a3 + a2 + a1 is divisible by 24.

Step-by-step solution

Understand the Structure of the Polynomial

Consider a polynomial of degree 4, given by \[ Q(x) = a_{4} x^{4} + a_{3} x^{3} + a_{2} x^{2} + a_{1} x \]where the coefficients \(a_{i}\) are integers. We need to show that \(Q(n)\) is divisible by 24 for all integers \(n \geq 0\) if certain conditions are met.

Check Divisibility by Smaller Numbers

Since 24 can be factored into \(24 = 2^3 \times 3\), it’s essential to verify divisibility by 8 (\(2^3\)) and by 3 separately.

Examine Divisibility by 8

Consider \(Q(n)\) modulo 8. For a polynomial to be divisible by 8, each coefficient must contribute to the total in a way that the sum is divisible by 8. Specifically:\[ Q(n) \mod 8 = a_4 n^4 + a_3 n^3 + a_2 n^2 + a_1 n \mod 8 \]For \( Q(n) \mod 8 = 0 \):The nonlinear terms \( n^4 \), \( n^3 \), \( n^2 \), and \( n \) modulo 8 must result in conditions on \(a_4\), \(a_3\), \(a_2\), and \(a_1\). We find that: \[ 2a_4 + a_3 \equiv 0 \mod 4 \]

Examine Divisibility by 3

Next, consider \( Q(n) \mod 3 \). Again, every term in \( Q(n)\) must be arranged such that their combined sum is divisible by 3. This results in:\[ a_4 + a_2 \equiv 0 \mod 3 \]Combined with step 3, we derive conditions for both modular divisions.

Combine Results for Overall Divisibility

Finally, combine conditions from the above steps to ensure the overall polynomial divisible by 24. We derive combined conditions:(i) \(2 a_{4} + a_{3} \equiv 0 \mod 4\) and check combining (ii) and (iii):(ii) \(a_{4} + a_{2} \equiv 0 \mod 12\), giving\(a_{3} \equiv 0 \mod 3 \)That results final condition: (iii) \( a_4 + a_3 + a_2 + a_1 \equiv 0 \mod 24\).

Conclusion of The Result

Therefore, \( Q(x) \) is divisible by 24 for all integers \( n \geq 0 \) if the conditions (i) \(2 a_{4}+a_{3} \equiv 0 \mod 4 \), (ii) \(a_{4}+a_{2} \equiv 0 \mod 12\), and (iii) \(a_{4}+a_{3}+a_{2}+a_{1} \equiv 0 \mod 24\) are all satisfied.

Key concepts

Integer Coefficients

In polynomial functions, the coefficients are the numbers placed in front of the variables. When we say these coefficients are integers, it means each coefficient is a whole number. This is important because calculations involving integers maintain their properties without introducing fractions or decimals.
For example, in the polynomial \(Q(x) = a_{4} x^{4} + a_{3} x^{3} + a_{2} x^{2} + a_{1} x \), every \(a_i\) value is an integer. This makes it easier to apply arithmetic rules and simplifies problems involving divisibility, as whole numbers are straightforward to manipulate.

Modular Arithmetic

Modular arithmetic involves working with the remainders of numbers after division. It's often called 'clock arithmetic' because it cycles back around after reaching a certain value called the modulus.
For this exercise, we looked at the polynomial \( Q(x) \) modulo different numbers like 8 and 3. When we say \( Q(x) \) modulo 8, we're interested in what the polynomial equals after dividing by 8 and looking at only the remainder.
For example, if a polynomial \( P(x) = 9x^2 + 6x + 7 \) and we look at it modulo 8:

• \(9x^2 \mod 8 = 1x^2 \)
• \(6x \mod 8 = 6x \)
• \(7 \mod 8 = 7 \)

So, \( P(x) \mod 8 = x^2 + 6x + 7 \). Using such calculations, we derived divisibility conditions for our original problem.

Divisibility Rules

Divisibility rules are shortcuts to determine whether one number divides another without performing full division. For this polynomial exercise, we used rules related to the number 24, influenced by its prime factorization, \(24 = 2^3 \times 3\).
This led us to check divisibility by 8 and by 3 separately. We established rules such as:

• For a number to be divisible by 8, specific parts of our polynomial had to consistently return a sum divisible by 8.
• Similarly, divisibility by 3 demanded certain terms of the polynomial to meet conditions ensuring the sum divided evenly by 3.

These rules helped derive conditions on the coefficients of the polynomial, demonstrating that each piece of our polynomial aligns with these simplifying checks.

Polynomial Functions

Polynomials are expressions involving variables raised to different powers, summed up with coefficients. For instance, \(Q(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x\) is a polynomial of degree 4, meaning the highest exponent in it is 4.
To test a polynomial for divisibility by 24, we examined its behavior under lower modulus values like 8 and 3, reflecting the prime factors of 24. We inspected:

• How the terms \(x^4, x^3, x^2\) and \(x\) combine under these moduli.
• Established coefficient conditions like \(2a_4 + a_3\) being divisible by 4.
• Ensured the combined rules collectively guarantee \(Q(x)\)'s outcome under the modulus 24.

Thus, we concluded that the polynomial function followed specific rules to ensure divisibility by complex numbers like 24, leveraging simpler modular arithmetic checks.