Let \(k\) be a positive integer. What is the quotient when \(x^{8k+3}+x^{8k+6}+x^{8k+9} + x^{8k+12}\) is divided by \((1+x^3)(1+x^6)\) ?
Polynomial Division: Finding the Quotient
We are asked to find the quotient when the polynomial \(P(x) = x^{8k+3}+x^{8k+6}+x^{8k+9} + x^{8k+12}\) is divided by the expression \(D(x) = (1+x^3)(1+x^6)\). Here, \(k\) is given as a positive integer.
Simplifying the Divisor Expression
First, let's expand the divisor \(D(x)\): \( D(x) = (1+x^3)(1+x^6) \) Using the distributive property (or FOIL method): \( D(x) = 1 \cdot 1 + 1 \cdot x^6 + x^3 \cdot 1 + x^3 \cdot x^6 \) \( D(x) = 1 + x^6 + x^3 + x^{3+6} \) \( D(x) = 1 + x^3 + x^6 + x^9 \) So, the divisor is \(1 + x^3 + x^6 + x^9\).
Simplifying the Dividend Polynomial
Now let's look at the dividend polynomial \(P(x)\): \( P(x) = x^{8k+3}+x^{8k+6}+x^{8k+9} + x^{8k+12} \) We can factor out the lowest power of \(x\), which is \(x^{8k+3}\): \( P(x) = x^{8k+3} \left( \frac{x^{8k+3}}{x^{8k+3}} + \frac{x^{8k+6}}{x^{8k+3}} + \frac{x^{8k+9}}{x^{8k+3}} + \frac{x^{8k+12}}{x^{8k+3}} \right) \) Using the exponent rule \(\frac{x^a}{x^b} = x^{a-b}\): \( P(x) = x^{8k+3} (x^{(8k+3)-(8k+3)} + x^{(8k+6)-(8k+3)} + x^{(8k+9)-(8k+3)} + x^{(8k+12)-(8k+3)}) \) \( P(x) = x^{8k+3} (x^0 + x^3 + x^6 + x^9) \) Since \(x^0 = 1\): \( P(x) = x^{8k+3} (1 + x^3 + x^6 + x^9) \)
Calculating the Quotient
The quotient \(Q(x)\) is obtained by dividing the dividend \(P(x)\) by the divisor \(D(x)\): \( Q(x) = \frac{P(x)}{D(x)} \) Substitute the simplified expressions for \(P(x)\) and \(D(x)\): \( Q(x) = \frac{x^{8k+3} (1 + x^3 + x^6 + x^9)}{1 + x^3 + x^6 + x^9} \) We can see that the term \((1 + x^3 + x^6 + x^9)\) appears in both the numerator and the denominator. Provided that this term is not zero, we can cancel it out: \( Q(x) = x^{8k+3} \)
Therefore, the quotient when \(x^{8k+3}+x^{8k+6}+x^{8k+9} + x^{8k+12}\) is divided by \((1+x^3)(1+x^6)\) is \(x^{8k+3}\).
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