In Exercises 41–50, subtract the polynomials. Assume that all variable exponents represent whole numbers.(17x³ − 5x² + 4x − 3) − (5x³ − 9x² − 8x + 11)

Verified step by step guidance

1. Identify the polynomials to be subtracted: \((17x^3 - 5x^2 + 4x - 3)\) 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. Rewrite the expression by combining the polynomials: \((17x^3 - 5x^2 + 4x - 3) + (-5x^3 + 9x^2 + 8x - 11)\).
4. Combine like terms: \((17x^3 - 5x^3) + (-5x^2 + 9x^2) + (4x + 8x) + (-3 - 11)\).
5. Simplify each group of like terms to find the resulting polynomial.

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables raised to whole number exponents, combined using addition, subtraction, and multiplication. Each term in a polynomial is made up of a coefficient and a variable part. Understanding the structure of polynomials is essential for performing operations like addition and subtraction.

Subtracting Polynomials

Subtracting polynomials involves distributing the negative sign across the terms of the polynomial being subtracted and then combining like terms. This process requires careful attention to the signs of each term to ensure accuracy. The result is a new polynomial that reflects the difference between the two original polynomials.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting coefficients of terms that have the same variable and exponent. This step is crucial when working with polynomials, as it allows for a more concise expression. For example, in the expression 3x² + 5x², the like terms can be combined to yield 8x².