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Chapter 6: Problem 445

Divide each polynomial by the monomial. $$\frac{42 x^{2}-14 x}{7}$$

Short Answer

Expert verified
6x^2 - 2x

Step by step solution

01

- Rewrite the Division

Express the division of the polynomial by the monomial as a single fraction. Rewrite the given expression: \[ \frac{42x^2 - 14x}{7} \].
02

- Split the Fraction

Separate the fraction into two parts, dividing each term of the polynomial by the monomial individually: \[ \frac{42x^2}{7} - \frac{14x}{7} \].
03

- Simplify Each Term

Simplify each fraction by dividing the coefficients and reducing powers of x if possible: \[ \frac{42x^2}{7} = 6x^2 \] and \[ \frac{14x}{7} = 2x \].
04

- Combine the Results

Combine the simplified terms to get the final result: \[ 6x^2 - 2x \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial
A polynomial is a mathematical expression consisting of variables, coefficients, and the operations of addition, subtraction, and multiplication. Each term in a polynomial is composed of a coefficient and a variable raised to a non-negative integer power, such as in the expression: \(42x^2 - 14x\). These can be added or subtracted together to form the polynomial.
Monomial
Unlike a polynomial, which has multiple terms, a monomial has only one term. It is of the form \(ax^n\), where \(a\) is a coefficient, and \(n\) is a non-negative integer. For example, in the expression \(7\), the term \(7\) is a monomial because it is a single term with no variables. When dividing polynomials by monomials, we treat each term in the polynomial separately against the monomial.
Fraction Simplification
Fraction simplification involves reducing fractions to their simplest form. When dividing polynomials by monomials, we can simplify each term of the polynomial by dividing the coefficients and reducing the variables' exponents if possible. For our example, simplify each term separately: \(\frac{42x^2}{7}\ = 6x^2\) and \(\frac{14x}{7}\ = 2x\). This makes the computation much simpler and the expression easier to manage.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It helps us understand how to perform operations like addition, subtraction, multiplication, and division. For instance, in polynomial division, using algebra, we break down complex expressions into simpler parts. Algebraic techniques help us divide polynomials by monomials, as we did in the exercise to get the results \(6x^2 - 2x\). Understanding algebra is crucial for simplifying and solving polynomial-related problems.

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Most popular questions from this chapter

Divide the monomials. $$\frac{65 a^{10} b^{8} c^{5}}{42 a^{7} b^{6} c^{8}}$$

Divide each polynomial by the monomial. $$\frac{412 z^{8}-48 z^{5}}{4 z^{3}}$$

Divide each polynomial by the monomial. $$\frac{20 b^{2}-12 b}{-4}$$

Divide each polynomial by the monomial. $$\left(63 m^{4}-42 m^{3}\right) \div\left(-7 m^{2}\right)$$

Simplify. (a) \(\frac{1}{q^{-10}}\) (b) \(\frac{1}{10^{-3}}\)
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