Problem 7 - Algebra for College Students

Question

Perform the indicated divisions of polynomials by monomials. $$ \frac{13 x^{3}-17 x^{2}+28 x}{-x} $$

Step-by-Step Solution

Answer

The division results in \(-13x^2 + 17x - 28\).

Step 1: Identify the Polynomials and the Monomial

In this expression, the polynomial is the numerator, which is \(13x^3 - 17x^2 + 28x\), and the monomial in the denominator is \(-x\). Our task is to perform the division of each term in the polynomial by the monomial \(-x\).

Step 2: Divide Each Term of the Polynomial by the Monomial

Divide each term of the polynomial by \(-x\):1. \(\frac{13x^3}{-x} = -13x^2\) (since \(x^3 \div x = x^{3-1} = x^2\))2. \(\frac{-17x^2}{-x} = 17x\) (since \(-x^2 \div -x = x^{2-1} = x\))3. \(\frac{28x}{-x} = -28\) (since \(x \div x = x^{1-1} = x^0 = 1\))

Step 3: Write the Resulting Expression

Combine the results from each division:The resulting expression is \(-13x^2 + 17x - 28\).

Key Concepts

Monomial

A monomial is a fundamental concept in algebra. It's a type of polynomial that contains only one term.
This means it consists of a coefficient, variables, and non-negative integer exponents, like \(x\) in our division problem.

• **Coefficient**: A number multiplying the variable(s), like \-1\ in \-x\.
• **Variable**: Often represented by letters, such as \(x\).
• **Exponent**: Represents the power to which a variable is raised, such as \( x^{3}\) in \(13 x^{3} \).

Understanding monomials is crucial as they are the building blocks for more complex polynomials.
They form the basis for many algebraic operations, including polynomial division.

Step-by-step solution

Breaking down a math problem into a step-by-step solution is incredibly helpful.
It turns an intimidating expression into smaller, manageable parts. Here's how to apply this to divide a polynomial by a monomial:

**Step 1:** Identify the polynomial and the monomial.
In this exercise, \(13x^3 - 17x^2 + 28x\) is our polynomial and \(-x\) is our monomial.

**Step 2:** Divide each term of the polynomial by the monomial.
For example, from \(13x^3\) we get \( \frac{13x^3}{-x} = -13x^2\). Do this for every term.

**Step 3:** Combine the results to write the final expression.
This gives us the simplified expression: \(-13x^2 + 17x - 28\).

Following step-by-step methods not only makes algebra easier but also more approachable for college students and others brushing up on their math skills.

Algebra for college students

Algebra is a core subject for college students pursuing science, technology, engineering, or mathematics.
Mastering it lays the groundwork for advanced studies. Key topics include polynomials, like monomials and polynomial divisions, to equip you with problem-solving skills.
Here are essential tips for acing college algebra:

• **Practice Regularly**: Familiarize yourself with exercises like polynomial division. It reinforces your understanding.
• **Understand the Underlying Concepts**: Don't just memorize formulas. Grasp what terms represent and why procedures work.
• **Use Study Groups**: Collaborative learning helps clarify complex problems. Exchange ideas and solutions.

Algebra fosters logical thinking and an analytical mindset, necessary for many professional fields. So, embrace these foundations to succeed in your career and academic pursuits.