Perform each division. See Examples 7 and 8. $(-4x^7-14x^6+10x^4-14x^2)/(-2x^2)$

Verified step by step guidance

1. Identify the division problem as dividing each term in the numerator by the denominator: \(\frac{-4x^7 - 14x^6 + 10x^4 - 14x^2}{-2x^2}\).
2. Rewrite the expression as separate fractions for each term: \(\frac{-4x^7}{-2x^2} + \frac{-14x^6}{-2x^2} + \frac{10x^4}{-2x^2} + \frac{-14x^2}{-2x^2}\).
3. Simplify the coefficients by dividing the numbers in the numerator by the number in the denominator for each term.
4. Simplify the variables by subtracting the exponents of \(x\) in the denominator from the exponents of \(x\) in the numerator for each term, using the rule \(\frac{x^a}{x^b} = x^{a-b}\).
5. Write the simplified terms together to form the final expression after division.

Key Concepts

Polynomial Division by a Monomial

Dividing a polynomial by a monomial involves dividing each term of the polynomial individually by the monomial. This simplifies the expression by reducing the degree of each term according to the division, making it easier to handle and interpret.

Laws of Exponents

When dividing terms with the same base, subtract the exponent of the divisor from the exponent of the dividend. For example, x^7 ÷ x^2 = x^(7-2) = x^5. This rule is essential for simplifying each term correctly during division.

Handling Negative Signs in Division

When dividing expressions with negative coefficients, apply the rule that dividing a negative by a negative yields a positive, while dividing a negative by a positive yields a negative. Correctly managing signs ensures the final simplified expression is accurate.