In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (5x²−7x−8)+(2x²−3x+7)−(x²−4x−3)

Verified step by step guidance

1. Combine all the polynomials by removing the parentheses. Be careful with the signs, especially for the subtraction of the third polynomial. The expression becomes: (5x^2 - 7x - 8) + (2x^2 - 3x + 7) - (x^2 - 4x - 3).
2. Distribute the negative sign across the third polynomial. This changes the signs of all terms in the third polynomial. The expression now becomes: 5x^2 - 7x - 8 + 2x^2 - 3x + 7 - x^2 + 4x + 3.
3. Group like terms together. Combine all the x^2 terms, all the x terms, and all the constant terms. The grouped expression is: (5x^2 + 2x^2 - x^2) + (-7x - 3x + 4x) + (-8 + 7 + 3).
4. Simplify each group by performing the addition or subtraction. For the x^2 terms, simplify 5x^2 + 2x^2 - x^2. For the x terms, simplify -7x - 3x + 4x. For the constants, simplify -8 + 7 + 3.
5. Write the resulting polynomial in standard form, which means arranging the terms in descending order of their degrees. Identify the degree of the polynomial by looking at the highest power of x in the final expression.

Key Concepts

Polynomial Addition and Subtraction

Polynomial addition and subtraction involve combining like terms from two or more polynomials. To perform these operations, you align the polynomials and add or subtract the coefficients of the terms with the same degree. This process is essential for simplifying expressions and finding the resulting polynomial.

Standard Form of a Polynomial

The standard form of a polynomial is when the terms are arranged in descending order of their degrees, starting from the highest degree to the lowest. This format makes it easier to identify the leading term and the degree of the polynomial, which is the highest exponent present in the expression.

Degree of a Polynomial

The degree of a polynomial is defined as the highest power of the variable in the polynomial expression. It provides important information about the polynomial's behavior, such as the number of roots and the end behavior of its graph. Understanding the degree is crucial for analyzing polynomial functions.