Polynomial Remainder Theorem Explained
This problem involves finding the remainder of a polynomial division using the Remainder Theorem. We are given information about the remainders when the polynomial \(p(x)\) is divided by simpler linear factors, \((x-1)\) and \((x-2)\). We need to determine the remainder when \(p(x)\) is divided by the product of these factors, \((x-1)(x-2)\).
Remainder Theorem Application
The Remainder Theorem states that if a polynomial \(p(x)\) is divided by a linear factor \((x-a)\), the remainder is \(p(a)\).
- We are given that when \(p(x)\) is divided by \((x-1)\), the remainder is 2. According to the Remainder Theorem, this means:
\(p(1) = 2\)
- We are also given that when \(p(x)\) is divided by \((x-2)\), the remainder is 1. This implies:
\(p(2) = 1\)
Remainder Form for Quadratic Divisor
When a polynomial \(p(x)\) is divided by another polynomial, the degree of the remainder must be less than the degree of the divisor.
- In this case, the divisor is \((x-1)(x-2)\), which is a quadratic polynomial (degree 2).
- Therefore, the remainder, let's call it \(R(x)\), must be a polynomial of degree less than 2. This means the remainder can be represented in the form \(ax + b\), where \(a\) and \(b\) are constants.
Polynomial Division Equation Setup
We can express the polynomial \(p(x)\) using the division algorithm:
\(p(x) = Q(x) \cdot (x-1)(x-2) + R(x)\)
Where \(Q(x)\) is the quotient and \(R(x)\) is the remainder. Substituting \(R(x) = ax + b\), we get:
\(p(x) = Q(x) \cdot (x-1)(x-2) + (ax + b)\)
Solving Remainder Coefficients
Now, we use the information we have about \(p(1)\) and \(p(2)\):
- Substitute \(x=1\) into the equation:
\(p(1) = Q(1) \cdot (1-1)(1-2) + (a(1) + b)\)
\(p(1) = Q(1) \cdot (0)(-1) + a + b\)
\(p(1) = 0 + a + b\)
Since we know \(p(1) = 2\), we have:
\(a + b = 2 \quad \quad (1)\)
- Substitute \(x=2\) into the equation:
\(p(2) = Q(2) \cdot (2-1)(2-2) + (a(2) + b)\)
\(p(2) = Q(2) \cdot (1)(0) + 2a + b\)
\(p(2) = 0 + 2a + b\)
Since we know \(p(2) = 1\), we have:
\(2a + b = 1 \quad \quad (2)\)
System of Linear Equations Solution
We now have a system of two linear equations with two variables, \(a\) and \(b\):
| Equation (1): | \(a + b = 2\) |
| Equation (2): | \(2a + b = 1\) |
To solve this system, we can subtract Equation (1) from Equation (2):
\((2a + b) - (a + b) = 1 - 2\)
\(2a + b - a - b = -1\)
\(a = -1\)
Now, substitute the value of \(a\) back into Equation (1):
\((-1) + b = 2\)
\(b = 2 + 1\)
\(b = 3\)
Final Remainder Calculation
We found the coefficients \(a = -1\) and \(b = 3\). The remainder \(R(x)\) was defined as \(ax + b\). Therefore, the remainder is:
\(R(x) = (-1)x + 3\)
\(R(x) = 3 - x\)
So, when the polynomial \(p(x)\) is divided by \((x-1)(x-2)\), the remainder is \(3-x\).
Similar Questions
- 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 ?
- 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 ?
