1. The remainder is zero when the polynomial is divided by \(x^2 + 1\).
2. The remainder is zero when the polynomial is divided by \(x^4 + 1\).
Which of the statements given above is/are correct?
Analyzing Polynomial Remainders
The question asks us to evaluate two statements concerning the remainders when the polynomial \(P(x) = x^{4k} + x^{4k+2} + x^{4k+4} + x^{4k+6}\) is divided by certain expressions.
Statement 1: Remainder when divided by \(x^2 + 1\)
To find the remainder when \(P(x)\) is divided by \(x^2 + 1\), we can use the property that if \(x^2 + 1 = 0\), then \(x^2 = -1\). We substitute \(x^2 = -1\) into the polynomial:
The polynomial is \(P(x) = x^{4k} + x^{4k+2} + x^{4k+4} + x^{4k+6}\).
Let's simplify each term using \(x^2 = -1\):
- \(x^{4k} = (x^2)^{2k} = (-1)^{2k}\)
- Since \(2k\) is always an even number, \((-1)^{2k} = 1\).
- \(x^{4k+2} = x^{4k} \cdot x^2 = (1) \cdot (-1) = -1\).
- \(x^{4k+4} = x^{4k} \cdot x^4 = (1) \cdot (x^2)^2 = (1) \cdot (-1)^2 = 1 \cdot 1 = 1\).
- \(x^{4k+6} = x^{4k} \cdot x^6 = (1) \cdot (x^2)^3 = (1) \cdot (-1)^3 = 1 \cdot (-1) = -1\).
Now, substitute these values back into the polynomial:
\(P(x)\) evaluated at \(x^2 = -1\) becomes \(1 + (-1) + 1 + (-1) = 0\).
Since the result is 0, the remainder is 0 when the polynomial is divided by \(x^2 + 1\). Therefore, statement 1 is correct.
Statement 2: Remainder when divided by \(x^4 + 1\)
To find the remainder when \(P(x)\) is divided by \(x^4 + 1\), we can use the property that if \(x^4 + 1 = 0\), then \(x^4 = -1\).
Alternatively, we can try to factor the polynomial \(P(x)\).
Let's factor \(P(x)\): \(P(x) = x^{4k} + x^{4k+2} + x^{4k+4} + x^{4k+6}\) Factor out the common term \(x^{4k}\): \(P(x) = x^{4k} (1 + x^2 + x^4 + x^6)\) Now, let's factor the expression inside the parenthesis: \(1 + x^2 + x^4 + x^6 = (1 + x^2) + x^4(1 + x^2)\) Factor out the common term \((1 + x^2)\): \(= (1 + x^2)(1 + x^4)\) So, the polynomial can be written as: \(P(x) = x^{4k} (1 + x^2) (1 + x^4)\)
From the factored form \(P(x) = x^{4k} (1 + x^2) (x^4 + 1)\), we can see that \((x^4 + 1)\) is a factor of \(P(x)\).
If an expression is a factor of a polynomial, the remainder upon division is 0.
Therefore, statement 2 is also correct.
Conclusion
Both statement 1 and statement 2 are correct.
The correct option is the one that states both statements are correct.
Important Questions from Polynomials
In the expansion of (x + 3) 3, the coefficient of x is:
If the degree of polynomial 9 x 5y 2z ris 15, then r = ?
The factorisation of x 2+ 11xy + 24y 2is:
The value of 16x 4+ 25y 2– 40x 2y at x = 5 and y = 2 is:
If the sum of the squares of the zeros of quadratic polynomial f(x) = x 2– 8x + k is 40, then find the value of k.
