Problem 96 - Elementary Algebra

Question

Perform the division: \(\frac{15 x^{3}-5 x^{2}+10 x}{5 x}\).

Step-by-Step Solution

Answer

Question: Given the expression \(\frac{15x^3 - 5x^2 + 10x}{5x}\), determine the result after dividing each term of the polynomial by the divisor. Answer: \(3x^2 - x + 2\)

Step 1: Divide the first term

Divide the first term \(15x^3\) by \(5x\): $$\frac{15x^3}{5x} = 3x^2$$.

Step 2: Divide the second term

Divide the second term \(-5x^2\) by \(5x\): $$\frac{-5x^2}{5x} = -x$$.

Step 3: Divide the third term

Divide the third term \(10x\) by \(5x\): $$\frac{10x}{5x} = 2$$.

Step 4: Combine the terms

Combine the terms from steps 1, 2, and 3 to find the final result: $$3x^2 -x + 2$$. The result of the division is: \(3x^2 - x + 2\).

Key Concepts

Monomial Division

Monomial division is a key aspect of dividing polynomials. It involves dividing a polynomial by a monomial, which is just a single term. This process can be simplified by dividing each term of the polynomial separately by the monomial.
For example, consider the expression \( \frac{15 x^{3}-5 x^{2}+10 x}{5 x} \). Here, our monomial is \( 5x \).

• To perform monomial division, take each term of the polynomial individually and divide it by the monomial \( 5x \).
• Make use of the law of exponents, which says: \( x^a / x^b = x^{a-b} \), to simplify the division of terms.

By applying these rules, the problem is broken down into manageable parts, allowing you to simplify each term separately before combining them for the final answer.

Simplifying Expressions

Simplifying expressions is a fundamental practice in algebra, and it's all about rewriting an expression in a simpler or more intuitive form. When we simplify an algebraic expression, we essentially aim to make it easily manageable without changing its value.
Consider the expression \( 15x^3 - 5x^2 + 10x \) divided by \( 5x \).

• First, handle each term separately to make the calculations straightforward. In the original problem, each term of the polynomial is divided by the monomial \( 5x \), which simplifies the expression.
• For example, \( \frac{15x^3}{5x} = 3x^2 \) simplifies because \( 15/5 = 3 \) and for variables, subtract the exponents: \( x^{3-1} = x^2 \).
• The final goal is to combine the results of these simpler calculations to obtain a new, more concise expression: \( 3x^2 - x + 2 \).

Remember, simplifying doesn't change the value of the expression, just the form.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are the building blocks of algebra, used to represent real-world scenarios and solve problems.
In our exercise, \( 15x^3 - 5x^2 + 10x \) is an algebraic expression. Here's how it can be understood and used:

• An algebraic expression can involve addition, subtraction, multiplication, and division of variables (like \( x \)) and constants (like \( 15, 5, 10 \)).
• Each term in an algebraic expression is separated by a plus \( + \) or minus \( - \) sign, making it easier to identify individual parts to operate on, such as during division or a simplification task.
• When dividing by a monomial, each term is addressed separately, which maintains the algebraic structure and aids in understanding the individual parts before looking at the whole.

Understanding these expressions helps in problem-solving and provides the tools needed to approach complex algebraic challenges.