In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (17x³−5x²+4x−3)−(5x³−9x²−8x+11)

검증된 단계별 안내

Start by distributing the subtraction sign across the second polynomial. This means you will change the sign of each term in the second polynomial: \((5x^3 - 9x^2 - 8x + 11)\) becomes \((-5x^3 + 9x^2 + 8x - 11)\).

Rewrite the expression by combining the first polynomial and the modified second polynomial: \((17x^3 - 5x^2 + 4x - 3) + (-5x^3 + 9x^2 + 8x - 11)\).

Group like terms together. Like terms are terms with the same variable raised to the same power: \((17x^3 - 5x^3) + (-5x^2 + 9x^2) + (4x + 8x) + (-3 - 11)\).

Simplify each group of like terms by performing the indicated addition or subtraction: Combine the coefficients of \(x^3\), \(x^2\), \(x\), and the constant terms.

Write the resulting polynomial in standard form, which means arranging the terms in descending order of their degree (highest power of \(x\) first). Then, identify the degree of the polynomial, which is the highest power of \(x\) in the polynomial.

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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 highest to 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 simplifies further analysis.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. It provides important information about the polynomial's behavior and its graph. For instance, in the polynomial 4x^3 - 2x + 5, the degree is 3, indicating that the highest exponent of x is 3, which influences the shape and number of roots of the polynomial.

In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (17x3−5x2+4x−3)−(5x3−9x2−8x+11)