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 terms in the numerator: \(-4x^7\), \(-14x^6\), \(10x^4\), and \(-14x^2\).
2. Identify the term in the denominator: \(-2x^2\).
3. Divide each term in the numerator by the term in the denominator separately.
4. For each division, subtract the exponents of \(x\) in the numerator and denominator: \(x^a / x^b = x^{a-b}\).
5. Simplify each resulting term to get the final expression.

Key Concepts

Polynomial Division

Polynomial division is the process of dividing one polynomial by another, similar to numerical long division. In this case, we divide the polynomial (-4x^7 - 14x^6 + 10x^4 - 14x^2) by the monomial (-2x^2). The result is obtained by dividing each term of the polynomial by the divisor, simplifying the expression step by step.

Simplifying Expressions

Simplifying expressions involves reducing complex algebraic expressions to their simplest form. This includes combining like terms and reducing coefficients. In the context of polynomial division, after dividing each term, it is essential to simplify the resulting expression to make it easier to interpret and work with.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, x^(-n) = 1/(x^n). In polynomial division, understanding how to handle negative coefficients and exponents is crucial, as it affects the signs and values of the resulting terms after division.