Problem 19 - Beginning and Intermediate Algebra

Question

Divide. Irene divides \(16 t^{3}-36 t^{2}+4 t\) by \(4 t\) and gets a quotient of \(4 t^{2}-9 t\). Is this correct? Why or why not?

Step-by-Step Solution

Answer

Irene's result is not correct because she missed the constant term in her quotient. The correct quotient is \(4t^{2} - 9t + 1\).

Step 1: Set up the division

Set up the division in the following way: \( \frac{16 t^{3}-36t^{2}+4t}{4t} \). We need to divide each term of the polynomial by the monomial.

Step 2: Divide the terms of the polynomial by the monomial

Now, divide each term of the polynomial by the monomial \(4t\): \( \frac{16 t^{3}}{4 t} - \frac{36t^{2}}{4t} + \frac{4t}{4t} \).

Step 3: Simplify the fractions

Simplify each term by cancelling out the common factors: \( \frac{16 t^{3}}{4 t} \Rightarrow \frac{16}{4} \cdot \frac{t^{3}}{t} = 4t^{2} \), \( \frac{-36 t^{2}}{4 t}\Rightarrow -\frac{36}{4} \cdot \frac{t^{2}}{t} = -9t \), and \( \frac{4 t}{4 t} \Rightarrow \frac{4}{4} \cdot \frac{t}{t} = 1 \).

Step 4: Write out the simplified result

Now, write out the simplified result of the division: \( 4t^{2} - 9t + 1 \).

Step 5: Compare the result with the given quotient

Compare the result \( 4t^{2} - 9t + 1 \) with the given quotient \( 4t^{2} - 9t \). They are not the same, so the given quotient is not correct. The correct quotient should be \( 4t^{2} - 9t + 1 \). Irene's result is not correct because she missed the constant term in her quotient. The correct quotient is \(4t^{2} - 9t + 1\).

Key Concepts

Monomial Division

Monomial division is a fundamental concept that allows you to divide a polynomial by a monomial, which is essentially a single-term expression. To perform monomial division, you must divide each term of the polynomial by the monomial separately.
For example, if you have \(16t^3 - 36t^2 + 4t\) and you want to divide it by \(4t\), you handle each term one by one:

• Divide \(16t^3\) by \(4t\)
• Divide \(-36t^2\) by \(4t\)
• Divide \(4t\) by \(4t\)

This step-by-step process ensures that each term is simplified correctly. If you skip any part or make a mistake in calculation, the entire division may be affected. Consistency and attention to detail are crucial during monomial division. Mistakes, such as missing terms, can lead to incorrect answers, as seen in Irene’s error of omitting the constant term in her quotient.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to its simplest form, eliminating unnecessary terms, and making it easier to work with. In our context, after dividing each term of the polynomial by the monomial, the next step is to simplify the fractions obtained.
To simplify, factor out the greatest common factor in both the numerator and the denominator, cancel out common terms, and reduce the expression where possible:

• For \(\frac{16t^3}{4t}\), simplify it to \(4t^2\)
• For \(\frac{-36t^2}{4t}\), simplify it to \(-9t\)
• For \(\frac{4t}{4t}\), simplify it to \(1\)

This simplification is essential to achieve the correct and final simplified expression, \(4t^2 - 9t + 1\). Without full simplification, significant aspects of the expression might be lost, leading to errors like missing terms.

Quotient in Algebra

In algebra, the quotient is the result of dividing one expression by another. Properly determining the quotient when dividing polynomials or monomials involves maintaining accuracy throughout each step. In the original exercise, the quotient \(4t^2 - 9t + 1\) represents the simplified result of dividing the polynomial by the monomial.
The significance of accurately finding the quotient includes:

• Verifying each step is correctly executed, ensuring the correct quotient comes out intact.
• Checking that all terms are accounted for, as omitting any can change the entire solution, as seen when Irene missed the "+1" term.

Ultimately, understanding how to properly calculate the quotient allows for improved problem-solving skills in algebra and ensures that your answers are precise and complete.