Understanding the Factor Theorem for Polynomials
The question asks us to determine which of the given polynomials has \((x+2)\) as a factor. We can use the Factor Theorem to solve this efficiently. The Factor Theorem states that for a polynomial \(P(x)\), \((x-a)\) is a factor of \(P(x)\) if and only if \(P(a) = 0\). In this problem, our potential factor is \((x+2)\), which can be written as \((x - (-2))\). Therefore, we need to find the polynomial \(P(x)\) for which \(P(-2) = 0\).
Evaluating Polynomials at \(x = -2\)
We will substitute \(x = -2\) into each of the given polynomial options to check if the result is zero.
Option 1: \(P(x) = x^5-4x^4-3x^3 + 8x^2 - 14x + 12\)
Let's calculate \(P(-2)\): \(P(-2) = (-2)^5 - 4(-2)^4 - 3(-2)^3 + 8(-2)^2 - 14(-2) + 12\) \(P(-2) = (-32) - 4(16) - 3(-8) + 8(4) - (-28) + 12\) \(P(-2) = -32 - 64 + 24 + 32 + 28 + 12\) Combine the negative terms: \(-32 - 64 = -96\) Combine the positive terms: \(24 + 32 + 28 + 12 = 96\) \(P(-2) = -96 + 96 = 0\) Since \(P(-2) = 0\), \((x+2)\) is a factor of this polynomial.
Option 2: \(P(x) = x^5 +4x^4-3x^3 + 8x^2 - 14x + 12\)
Let's calculate \(P(-2)\): \(P(-2) = (-2)^5 + 4(-2)^4 - 3(-2)^3 + 8(-2)^2 - 14(-2) + 12\) \(P(-2) = (-32) + 4(16) - 3(-8) + 8(4) - (-28) + 12\) \(P(-2) = -32 + 64 + 24 + 32 + 28 + 12\) Combine the negative terms: \(-32\) Combine the positive terms: \(64 + 24 + 32 + 28 + 12 = 160\) \(P(-2) = -32 + 160 = 128\) Since \(P(-2) \neq 0\), \((x+2)\) is not a factor of this polynomial.
Option 3: \(P(x) = x^5-4x^4 + 3x^3 + 8x^2 - 14x + 12\)
Let's calculate \(P(-2)\): \(P(-2) = (-2)^5 - 4(-2)^4 + 3(-2)^3 + 8(-2)^2 - 14(-2) + 12\) \(P(-2) = (-32) - 4(16) + 3(-8) + 8(4) - (-28) + 12\) \(P(-2) = -32 - 64 - 24 + 32 + 28 + 12\) Combine the negative terms: \(-32 - 64 - 24 = -120\) Combine the positive terms: \(32 + 28 + 12 = 72\) \(P(-2) = -120 + 72 = -48\) Since \(P(-2) \neq 0\), \((x+2)\) is not a factor of this polynomial.
Option 4: \(P(x) = x^5-4x^4-3x^3 + 8x^2 + 14x + 12\)
Let's calculate \(P(-2)\): \(P(-2) = (-2)^5 - 4(-2)^4 - 3(-2)^3 + 8(-2)^2 + 14(-2) + 12\) \(P(-2) = (-32) - 4(16) - 3(-8) + 8(4) + (-28) + 12\) \(P(-2) = -32 - 64 + 24 + 32 - 28 + 12\) Combine the negative terms: \(-32 - 64 - 28 = -124\) Combine the positive terms: \(24 + 32 + 12 = 68\) \(P(-2) = -124 + 68 = -56\) Since \(P(-2) \neq 0\), \((x+2)\) is not a factor of this polynomial.
Conclusion
Based on the Factor Theorem and our calculations, only the first polynomial, \(x^5-4x^4-3x^3 + 8x^2 - 14x + 12\), results in \(P(-2) = 0\). Therefore, \((x+2)\) is a factor of this polynomial.
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 \(x^2 + 1\)?
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 ?
- 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? - 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 ?
