In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (17x^3−5x^2+4x−3)−(5x^3−9x^2−8x+11)

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1. Identify the polynomials involved in the subtraction: \((17x^3 - 5x^2 + 4x - 3)\) and \((5x^3 - 9x^2 - 8x + 11)\).
2. Distribute the negative sign across the second polynomial: \(-(5x^3 - 9x^2 - 8x + 11) = -5x^3 + 9x^2 + 8x - 11\).
3. Combine like terms from both polynomials: \((17x^3 - 5x^2 + 4x - 3) + (-5x^3 + 9x^2 + 8x - 11)\).
4. Add or subtract the coefficients of like terms: \((17x^3 - 5x^3) + (-5x^2 + 9x^2) + (4x + 8x) + (-3 - 11)\).
5. Write the resulting polynomial in standard form, arranging terms in descending order of degree, and determine the degree of the polynomial.

Key Concepts

Polynomial Operations

Polynomial operations involve adding, subtracting, multiplying, or dividing polynomials. In this case, we are focusing on subtraction, which requires distributing the negative sign across the second polynomial and combining like terms. Understanding how to manipulate polynomials is essential for solving problems involving them.

Standard Form of a Polynomial

The standard form of a polynomial is when the terms are arranged in descending order of their degrees, from the highest to the lowest. For example, a polynomial like 3x^2 + 2x + 1 is in standard form. Writing the resulting polynomial in standard form helps in easily identifying its degree and understanding its behavior.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable in the polynomial. It provides insight into the polynomial's behavior, such as the number of roots and the end behavior of its graph. For instance, in the polynomial 4x^3 - 2x + 1, the degree is 3, indicating it is a cubic polynomial.