Question

Find the unknown factor. $$44 n^{3}+33 n^{6}=11 n^{3}(?)$$

Short answer

)\)? Answer: The unknown factor is \(4 + 3n^3\).

Step-by-step solution

Identifying the common factor

First, observe that the given expression has a common factor of \(11n^3\). Step 2: Divide the expression by the common factor

Dividing the expression

Divide the given expression by \(11n^3\) to find the unknown factor: $$\frac{44 n^{3} + 33 n^{6}}{11 n^{3}}$$ Step 3: Simplify the expression

Simplifying the expression

Simplify the expression by dividing each term by \(11n^3\): $$\frac{44n^3}{11n^3} + \frac{33n^6}{11n^3}$$ Step 4: Compute the result

Calculating the result

Perform the division on each term: $$4 + 3n^3$$ Step 5: Write down the answer

Writing down the answer

The unknown factor is: $$(?) = 4 + 3n^3$$

Key concepts

Common Factor

A common factor is a factor that is shared by all terms in an expression. In this exercise, we can see that the two terms in the expression are \(44n^3\) and \(33n^6\). Both these terms share the factor \(11n^3\). This means that \(11n^3\) is the largest expression that can be factored out from each term.Finding the common factor is crucial because it simplifies the entire problem significantly. It allows us to break down complex expressions into simpler, more manageable parts. By extracting the common factor, we can rewrite the expression as a product of this factor and another polynomial, making computations easier and clearer.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. In our example, after identifying that \(11n^3\) is a common factor of both \(44n^3\) and \(33n^6\), the next logical step is division.By dividing each term by the common factor \(11n^3\) (i.e., \(\frac{44n^3}{11n^3} + \frac{33n^6}{11n^3}\)), we simplify the expression to \(4 + 3n^3\). Simplifying helps in achieving an expression that is easier to work with and understand. It removes any complications due to large numbers or higher powers, offering a clean result.This simplification is a vital skill not only in learning algebra but also in solving real-world problems more efficiently.

Polynomial Division

Polynomial division consists of dividing each term of a polynomial by a common divisor. After determining the common factor \(11n^3\), the task simplifies to performing the division on each term separately.For the polynomial \(44n^3 + 33n^6\), each term is divided by \(11n^3\):

• \(\frac{44n^3}{11n^3} = 4\)
• \(\frac{33n^6}{11n^3} = 3n^3\)

Thus, the polynomial \(44n^3 + 33n^6\) simplifies to \(11n^3(4 + 3n^3)\).Polynomial division, though resembling numerical division, involves considering powers of variables and coefficients carefully. It is key in expressing a polynomial as a product of a simpler polynomial and another expression, simplifying problem-solving.