Problem 3 - Beginning and Intermediate Algebra

Question

Explain, in your own words, how to divide a polynomial by a monomial.

Step-by-Step Solution

Answer

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial, then simplify the fractions and combine the simplified terms. For example, when dividing the polynomial \(4x^3 - 2x^2 + 8x\) by the monomial \(2x\), divide each term of the polynomial by \(2x\) and simplify: \( \frac{4x^3}{2x} = 2x^2\), \( \frac{-2x^2}{2x} = -x \), \( \frac{8x}{2x} = 4 \). Combine the simplified terms to get the final result: \(2x^2 - x + 4\).

Step 1: Understand the terms

A polynomial is an algebraic expression consisting of one or more terms, where each term has a coefficient and a variable with a non-negative exponent. For example, \(3x^2 + 7x - 4\) is a polynomial with three terms. A monomial is an algebraic expression with only one term, like \(2x\), \(5y^3\), or \(-3z^2\). Dividing a polynomial by a monomial means dividing each term of the polynomial by the monomial.

Step 2: Choose an example to demonstrate division

Let's take the example of dividing the polynomial \(4x^3 - 2x^2 + 8x\) by the monomial \(2x\).

Step 3: Divide each term of the polynomial by the monomial

Divide the first term of the polynomial \(4x^3\) by the monomial \(2x\): \( \frac{4x^3}{2x} \) Similarly, divide the second term of the polynomial \(-2x^2\) by the monomial \(2x\): \( \frac{-2x^2}{2x} \) Lastly, divide the third term of the polynomial \(8x\) by the monomial \(2x\): \( \frac{8x}{2x} \)

Step 4: Simplify the fractions

Now, let's simplify the fractions obtained from dividing each term: \( \frac{4x^3}{2x} = 2x^2 \) \( \frac{-2x^2}{2x} = -x \) \( \frac{8x}{2x} = 4 \)

Step 5: Combine the simplified terms

Combine the simplified expressions to get the final result: \(2x^2 - x + 4\) So, the result of dividing the polynomial \(4x^3 - 2x^2 + 8x\) by the monomial \(2x\) is \(2x^2 - x + 4\).