Problem 34 - Algebra 2

Question

Find values of \(k\) so that each remainder is \(3 .\) $$ \left(x^{2}-x+k\right) \div(x-1) $$

Step-by-Step Solution

Answer

The value of \(k\) is 3.

Step 1: Understand the Problem

The problem involves dividing a quadratic polynomial by a linear polynomial, and we need to find the values of \(k\) such that the remainder is a specific value (3 in this case).

Step 2: Identify the Division Algorithm

The division of polynomials can be expressed by the polynomial division algorithm which says: \(f(x) = (x - 1)q(x) + r\), where \(f(x) = x^2 - x + k\), \(q(x)\) is the quotient, and \(r\) is the remainder we seek to be 3.

Step 3: Apply Polynomial Remainder Theorem

According to the Polynomial Remainder Theorem, the remainder when a polynomial \(f(x)\) is divided by \(x - a\) is \(f(a)\). Here, \(x - 1\) means \(a = 1\). So, we need to find \(f(1)\).

Step 4: Substitute into the Polynomial

Substitute \(x = 1\) into \(f(x) = x^2 - x + k\) to find the remainder: \[ f(1) = 1^2 - 1 + k = 0 + k = k \]

Step 5: Set Remainder to 3

We want the remainder \(r\) to be 3. Thus, we set \(k = 3\).

Step 6: Conclusion

Therefore, the value of \(k\) which makes the remainder 3 is \(k = 3\).

Key Concepts

Polynomial Remainder Theorem

The Polynomial Remainder Theorem is a simplification tool in polynomial algebra. It provides a way to evaluate a polynomial's value, which is essential when dividing by a linear divisor. The theorem states: If a polynomial \( f(x) \) is divided by \( x-a \), the remainder of this division is \( f(a) \). To use this theorem, simply substitute \( a \) into the polynomial function. In our problem, we divide \( f(x) = x^2 - x + k \) by \( x-1 \), so \( a = 1 \). By applying the theorem, we substitute \( x = 1 \) into the polynomial to calculate the remainder directly, making these calculations quick and efficient.

Polynomial Division Algorithm

The Polynomial Division Algorithm resembles long division with numbers but applies to polynomials. It allows us to express the division of any polynomial \( f(x) \) by a divisor \( d(x) \) in the form:

• \( f(x) = d(x) \cdot q(x) + r \)

Here, \( q(x) \) represents the quotient polynomial and \( r \) is the remainder. For our exercise, we have a quadratic polynomial \( f(x) = x^2 - x + k \) divided by a linear polynomial \( x-1 \). With the desired remainder being 3, the polynomial division algorithm precisely outlines how polynomials can be broken down into simpler components, capturing the essential parts: the quotient and remainder.

Quotient and Remainder

In polynomial division, manipulating the terms in \( f(x) \) gives us crucial results, specifically the quotient and the remainder. The quotient \( q(x) \) is obtained by dividing the leading term of \( f(x) \) by that of \( d(x) \). Meanwhile, the remainder is derived from the remaining terms when the division stops. Critical for our problem, the remainder \( r \) needs to equal 3. The relationship:

• \( f(x) = (x-1) \cdot q(x) + r \)

allows us to easily plug in the remainder \( r \) into the equation and find the right \( k \) to match this condition.

Quadratic Polynomial

Quadratic polynomials are a class of polynomials characterized by their degree of two. They take the general form:

• \( q(x) = ax^2 + bx + c \)

where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). In our specific exercise, \( f(x) = x^2 - x + k \) is a quadratic polynomial. Here, \( a = 1 \), \( b = -1 \), and \( c = k \). Understanding the structure of these polynomials simplifies the manipulation during division. Quadratics can express a range of functions and are pivotal in both calculations and graph representation. Our task is simplified by leveraging their predictable pattern and structure.