Problem 58 - Algebra 2

Question

Critical Thinking Explain why a polynomial of degree \(n\) , divided by a polynomial of degree \(1,\) yields a quotient of degree \(n-1\) and a remainder that is a constant.

Step-by-Step Solution

Answer

Dividing a polynomial of degree n by a polynomial of degree 1 yields a quotient of degree n-1 because each subtraction of the linear polynomial reduces the degree by 1. The remainder is a constant because the division stops when the remainder has a degree lower than 1.

Step 1: Understanding Polynomial Division

When you divide a polynomial of degree n by a polynomial of degree 1, you are essentially asking how many times the divisor (a linear factor), which has a degree of 1, can be 'contained' within the dividend (the degree n polynomial) in terms of polynomial powers. Division is a form of repeated subtraction.

Step 2: Dividing by a Degree 1 Polynomial (Linear Factor)

Consider the divisor polynomial to be of the form ax + b where a and b are constants. Division by this polynomial is similar to long division, where at each stage, you subtract multiples of ax + b from the dividend polynomial. You always match the highest degree term of the divisor to the highest degree term of the dividend.

Step 3: Determining the Degree of the Quotient

The degree of the quotient is determined by the difference in degrees between the dividend and the divisor, which is n - 1. This is because when you subtract multiples of the linear factor from the highest degree term of the polynomial, you decrease its degree by 1.

Step 4: Understanding the Remainder

The process of division will continue until the degree of what remains is less than the degree of the divisor. Since the divisor is of degree 1, the remainder must be of degree 0, which implies that the remainder is a constant.

Key Concepts

Polynomial Degree

When discussing polynomials, the degree is a fundamental concept. It tells you the highest power of the variable in the polynomial. For example, in the polynomial \(3x^4 - 2x^3 + x - 5\), the highest power of \(x\) is 4; hence it is a fourth-degree polynomial.

The degree of a polynomial is significant because it tells us about the behavior of the graph of the polynomial, the possible number of roots, and the impact of each term when performing operations such as division.

• The degree predicts the graph's shape: as the degree increases, the number of turns on the graph can increase.
• It tells us about the potential number of x-intercepts or roots that the polynomial could have.
• During division, the highest degree terms have the most influence on the quotient.

By grasping the polynomial degree, you pave the way for understanding why polynomial division behaves as it does.

Linear Factor

A linear factor is a polynomial of degree 1, often represented in the form \(ax + b\), where \(a\) and \(b\) are constants, and \(a \eq 0\).

• The linear factor is crucial to dividing polynomials as it serves as the divisor in the polynomial long division.
• Each linear factor corresponds to a root of the polynomial. When \(ax + b = 0\), then \(x = -\frac{b}{a}\) is a root.
• Division by a linear factor simplifies the polynomial by reducing its degree by 1 for each division.

Understanding linear factors is not just about the mechanics of division; it is also about gaining insight into the structure and roots of polynomials.

Quotient and Remainder Theorem

The Quotient and Remainder Theorem is a statement about what happens when you divide a polynomial by a divisor of the form \(ax + b\). According to the theorem, when you divide any polynomial by a linear polynomial, you get a quotient and possibly a remainder. The form of the remainder depends on the degree of the divisor.

In our case, the divisor is of degree 1, so the remainder is of degree 0, meaning it is a constant. Why is that? As you divide and reduce the polynomial, each subtraction of a multiple of the divisor lowers the degree until no terms of a higher degree than the divisor remain. Thus, this theorem assures that when dividing by a linear factor, the remainder will always be a polynomial of lower degree—in this case, a constant.

Polynomial Division Steps

Working through polynomial division is akin to long division with numbers, but some steps are unique to polynomials. Let's briefly outline these steps:

Identify and Arrange the Dividend and Divisor

Ensure that the dividend and divisor polynomials are written in descending order of degrees.

Divide the Leading Terms

Find the quotient of the highest degree terms. This will become the first term of your answer.

Subtract and Bring Down

Multiply your answer by the divisor and subtract from the dividend. Bring down the next term.

Repeat the Process

Continue this process of dividing leading terms, subtracting, and bringing down subsequent terms until the degree of the remaining term is less than the degree of the divisor.

Identify the Remainder

Once the process is complete, any remaining polynomial of lower degree than the divisor is your remainder.
By understanding and practicing these steps, dividing polynomials becomes a manageable and logical process.