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.
Similar Questions
- Let \(p(x)\) be a polynomial. When \(p(x)\) is divided by \((x-1)\), it leaves 2 as the remainder. When \(p(x)\) is divided by \((x-2)\), it leaves 1 as the remainder. What is the remainder when \(p(x)\) is divided by \((x - 1)(x-2)\)?
- What is the remainder when \(x^6\) is divided by \(x^2 + 1\)?
- \((x+2)\) is a factor of which one of the following?
What is the HCF of the polynomials x⁸ + x⁴ + 1 and x⁴ + x² + 1?
- If 2 is a zero of the polynomial \(p(x) = x^3 + 3x^2 - 6x - a\), then what is the sum of the squares of the other zeros of the polynomial?
- Suppose \(p(x) = x^4 + a_3x^3 + a_2x^2 + a_1x + a_0\) and \(q(x) = x^4 + b_3x^3 + b_2x^2 + b_1x + b_0\) are the polynomials. If \(\alpha, \beta, \gamma, \delta\) are zeros of \(p(x)\) and \(\alpha, \beta, \gamma, \lambda\) are zeros of \(q(x)\), then what is \(\frac{p(x) - q(x)}{(x - \alpha) (x - \beta) (x - \gamma)}\) equal to ?
- If \(x^3 + px^2 + qx + r\) is an integer for all integral values of \(x\), then consider the following statements :
I. \(p\) must be an integer
II. \(q\) must be an integer
III. \(r\) must be an integer
Which of the statements given above is/are correct? - Which of the following expressions can divide both the polynomials \(x^3+2x^2-5x+2\) and \(x^3+4x^2+x-6\) exactly ?
I. \(x-1\)
II. \(x+1\)
III. \(x+2\)
Select the correct answer using the code given below : If x² - 5x + 4 is a factor of x⁴ - px² + q, then what are the values of p and q respectively?
Important Questions from Polynomials
If y 2= y + 7, then what is the value of y 3?
Factorize x 2- y 2- 9z 2+ 6yz
If one of the zeros of the polynomial x 3+ ax 2+ bx + c is - 1, then the product of other two zeros is equal to :
If a(a + b + c) 2 = 1792; b(a + b + c) 2 = 1536; c(a + b + c) 2 = 768, then what will be the value of b?
If x = 3 so, what is the value of x 2 + 2x + 5 ?
