Problem 101 - Algebra for College Students

Question

Discussion. On a test a student divided \(3 x^{3}-5 x^{2}-3 x+7\) by \(x-3\) and got a quotient of \(3 x^{2}+4 x\) and remainder \(9 x+7 .\) Verify that the divisor times the quotient plus the remainder is equal to the dividend. Why was the student's answer incorrect?

Step-by-Step Solution

Answer

The student's quotient and remainder did not follow correct polynomial division steps since the remainder should be a constant.

Step 1 - Write the Given Information

The problem states that the student divided the polynomial \(3x^3 - 5x^2 - 3x + 7\) by \(x - 3\) and obtained a quotient of \(3x^2 + 4x\) and a remainder of \(9x + 7\).

Step 2 - Verify the Polynomial Division

To verify the student's result, we need to check if the formula \( \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} \) holds true. Substitute the given values: \[3x^3 - 5x^2 - 3x + 7 = (x-3)(3x^2 + 4x) + (9x + 7).\]

Step 3 - Expand the Product of Divisor and Quotient

Calculate the product: \[(x - 3)(3x^2 + 4x).\] Multiply each term inside the parentheses: \[x \times 3x^2 + x \times 4x - 3 \times 3x^2 - 3 \times 4x.\] Simplify to get: \[3x^3 + 4x^2 - 9x^2 - 12x = 3x^3 - 5x^2 - 12x.\]

Step 4 - Add the Remainder

Add the remainder \(9x + 7\) to the result: \[3x^3 - 5x^2 - 12x + 9x + 7.\] Combine like terms: \[3x^3 - 5x^2 - 3x + 7.\]

Step 5 - Compare with the Dividend

Now, compare the expression obtained with the given dividend: \(3x^3 - 5x^2 - 3x + 7.\) The expressions match, so the arithmetic seems correct. However, the student's quotient and remainder combination is inconsistent with polynomial division rules.

Step 6 - Explain the Mistake

The student's error likely lies in the incorrect remainder: when dividing \(3x^3 - 5x^2 - 3x + 7\) by \(x - 3\), the true remainder should be a constant term, not a first-degree polynomial (i.e., the student did not correctly perform the polynomial division process). Consequently, the resulting quotient and remainder need to be recalculated correctly according to polynomial division steps.

Key Concepts

Polynomial Arithmetic

Polynomial arithmetic involves performing operations such as addition, subtraction, multiplication, and division on polynomials. Polynomials are algebraic expressions consisting of variables, coefficients, and exponents. In this problem, the polynomial given is of degree 3, meaning its highest exponent is 3. When dealing with polynomial division, we look at how many times a divisor (in this case, a polynomial of lower degree) fits into a dividend (the polynomial being divided). The goal is to find a quotient (result of the division) and possibly a remainder.

To divide polynomials, perform long division similar to numerical long division. You divide the leading term (highest degree) of the dividend by the leading term of the divisor, multiply the whole divisor by this result, and subtract it from the dividend to find the next term. This process repeats until the remainder is less than the degree of the divisor.

Remainder Theorem

The Remainder Theorem is a useful concept in polynomial arithmetic. It states that when a polynomial, say \( f(x) \), is divided by a linear divisor of the form \( x - c \), the remainder of this division is simply \( f(c) \). This means if you plug \( c \) into the polynomial, the result will be the remainder. The key insight here is to understand the nature of the remainder. When dividing by a polynomial of the form \( x - 3 \), the remainder should be a constant value. It shouldn't have any variable terms like \( x \) or \( x^2 \).

In the given problem, the error lies in having a remainder of \( 9x + 7 \), which is a linearly-dependent polynomial and not a constant. Correctly applying the Remainder Theorem reveals whether the work adheres to polynomial division principles.

Algebraic Verification

Algebraic verification is the process of using algebraic identities and operations to confirm the correctness of a mathematical outcome. For polynomial division, we verify the correctness of the result by ensuring that combining the divisor, quotient, and remainder reconstructs the original polynomial. The formula to check this is given by:
\(\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} \).
In the given problem, substituting the given values correctly recreated the original polynomial. However, algebraic verification also depends on correct polynomial division principles, ensuring the remainder is of a lower degree than the divisor.
When steps show inconsistencies, as seen with the student's linear remainder, it indicates errors in the polynomial division steps need correcting. Revisiting and properly recalculating the steps ensures an accurate quotient and remainder.