If the polynomial 6x4 + 8x3 + 17x2 + 21x + 7 is divided by another polynomial 3x2 + 4x +1, the remainder comes out to be ax + b, find a and b.
a = 1; b = 2
Polynomial Division Explained: Finding Remainder ax + b
We are given a polynomial \( P(x) = 6x^4 + 8x^3 + 17x^2 + 21x + 7 \) which is divided by another polynomial \( D(x) = 3x^2 + 4x + 1 \). We are told the remainder of this division is in the form \( ax + b \), and we need to find the values of \( a \) and \( b \).
To find the remainder, we perform polynomial long division.
The process of polynomial long division involves dividing the term with the highest power in the dividend by the term with the highest power in the divisor, multiplying the result by the entire divisor, and subtracting this product from the dividend. We repeat this process until the degree of the remainder is less than the degree of the divisor.
Let's perform the division step-by-step:
- Divide the leading term of \( P(x) \) (\( 6x^4 \)) by the leading term of \( D(x) \) (\( 3x^2 \)). \( \frac{6x^4}{3x^2} = 2x^2 \)
- Multiply the result (\( 2x^2 \)) by the divisor \( D(x) = 3x^2 + 4x + 1 \). \( 2x^2(3x^2 + 4x + 1) = 6x^4 + 8x^3 + 2x^2 \)
- Subtract this result from the dividend \( P(x) \). \( (6x^4 + 8x^3 + 17x^2 + 21x + 7) - (6x^4 + 8x^3 + 2x^2) \) \( = 6x^4 - 6x^4 + 8x^3 - 8x^3 + 17x^2 - 2x^2 + 21x + 7 \) \( = 15x^2 + 21x + 7 \)
- Now, consider the new polynomial \( 15x^2 + 21x + 7 \) as the dividend. Divide its leading term (\( 15x^2 \)) by the leading term of the divisor (\( 3x^2 \)). \( \frac{15x^2}{3x^2} = 5 \)
- Multiply this result (\( 5 \)) by the divisor \( D(x) = 3x^2 + 4x + 1 \). \( 5(3x^2 + 4x + 1) = 15x^2 + 20x + 5 \)
- Subtract this result from the current polynomial \( 15x^2 + 21x + 7 \). \( (15x^2 + 21x + 7) - (15x^2 + 20x + 5) \) \( = 15x^2 - 15x^2 + 21x - 20x + 7 - 5 \) \( = x + 2 \)
The resulting polynomial is \( x + 2 \). The degree of \( x + 2 \) is 1, which is less than the degree of the divisor \( 3x^2 + 4x + 1 \) (which is 2). Therefore, \( x + 2 \) is the remainder of the division.
We are given that the remainder is in the form \( ax + b \). Comparing our remainder \( x + 2 \) with \( ax + b \):
\( x + 2 = ax + b \)
By comparing the coefficients of \( x \) and the constant terms, we can find the values of \( a \) and \( b \):
- Coefficient of \( x \): The coefficient of \( x \) in \( x + 2 \) is 1, and in \( ax + b \) is \( a \). So, \( a = 1 \).
- Constant term: The constant term in \( x + 2 \) is 2, and in \( ax + b \) is \( b \). So, \( b = 2 \).
Thus, the values are \( a = 1 \) and \( b = 2 \).
This matches one of the given options.
Revision Table: Key Concepts in Polynomials
| Term | Definition |
|---|---|
| Polynomial | An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. |
| Dividend | The polynomial being divided. In \( P(x) / D(x) \), \( P(x) \) is the dividend. |
| Divisor | The polynomial by which the dividend is divided. In \( P(x) / D(x) \), \( D(x) \) is the divisor. |
| Quotient | The result of the division (excluding the remainder). |
| Remainder | The polynomial left over after division, whose degree is less than the divisor's degree. \( P(x) = D(x) \times Q(x) + R(x) \) where \( Q(x) \) is the quotient and \( R(x) \) is the remainder. |
| Degree of a Polynomial | The highest power of the variable in the polynomial. |
Additional Information: Understanding Polynomial Remainder
The remainder from polynomial division is important in algebra. The Remainder Theorem, for example, states that if a polynomial \( P(x) \) is divided by a linear factor \( (x - c) \), the remainder is \( P(c) \).
In this problem, the divisor is a quadratic, not linear, so the Remainder Theorem in its basic form is not directly applicable for finding the remainder's coefficients, but the concept of a remainder with a degree less than the divisor is fundamental.
Polynomial long division is a systematic way to divide polynomials, analogous to numerical long division. It helps in factoring polynomials, finding roots, and simplifying rational expressions. When the remainder is zero, it means the divisor is a factor of the dividend.
The coefficients of the remainder polynomial, like \( a \) and \( b \) in \( ax + b \), are uniquely determined by the division process.
Important Questions from Polynomials
If (x + y) 3+ 8 (x - y) 3= (3x + Ay) (3x 2+ Bxy + Cy 2), then the value of A + B + C is:
Given that x 8- 34x 4+ 1 = 0, x > 0. What is the value of (x 3+ x -3 )?
If \(x - \frac 3 x = 6,\; x \ne 0,\) then the value of \(\frac {x^4 - \frac {27}{x^2}}{x^2 - 3x - 3}\) is:
If \(x\left(3 - \frac 2 x\right) = \frac 3 x,\) then the value of \(x^3 - \frac 1 {x^3}\) is equal to:
The coefficient of x in (x – 3y) 3is:
