I. \(x-1\)
II. \(x+1\)
III. \(x+2\)
Select the correct answer using the code given below :
I only
To determine which expressions can divide both polynomials \( x^3 + 2x^2 - 5x + 2 \) and \( x^3 + 4x^2 + x - 6 \) exactly, we need to check each expression one-by-one:
Step 1: Check the divisibility by \( x-1 \)
We can determine if \( x-1 \) is a factor of a polynomial by substituting \( x = 1 \) into the polynomial and checking if it equals zero.
Substituting \( x = 1 \) into the first polynomial:
\(x^3 + 2x^2 - 5x + 2 = 1^3 + 2(1)^2 - 5(1) + 2 = 1 + 2 - 5 + 2 = 0\)
The first polynomial is divisible by \( x-1 \).
Substituting \( x = 1 \) into the second polynomial:
\(x^3 + 4x^2 + x - 6 = 1^3 + 4(1)^2 + 1 - 6 = 1 + 4 + 1 - 6 = 0\)
The second polynomial is also divisible by \( x-1 \).
Therefore, \( x-1 \) is a common factor.
Step 2: Check the divisibility by \( x+1 \)
Substitute \( x = -1 \) into the first polynomial:
\(x^3 + 2x^2 - 5x + 2 = (-1)^3 + 2(-1)^2 - 5(-1) + 2 = -1 + 2 + 5 + 2 = 8 \neq 0\)
The first polynomial is not divisible by \( x+1 \).
Since \( x+1 \) is not a factor of the first polynomial, it cannot be a common factor.
Step 3: Check the divisibility by \( x+2 \)
Substitute \( x = -2 \) into the first polynomial:
\(x^3 + 2x^2 - 5x + 2 = (-2)^3 + 2(-2)^2 - 5(-2) + 2 = -8 + 8 + 10 + 2 = 12 \neq 0\)
The first polynomial is not divisible by \( x+2 \).
Since \( x+2 \) is not a factor of the first polynomial, it cannot be a common factor.
Conclusion:
The only expression that divides both polynomials exactly is \( x-1 \), which corresponds to the correct answer: I only.
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?
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- Consider the following in respect of the polynomial \(x^{4k} + x^{4k+2} + x^{4k+4} + x^{4k+6}\):
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? - 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? 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 ?
