Problem 63 - Algebra and Trigonometry

Question

In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. $$ \left(x^{3}+7 x y-5 y^{2}\right)-\left(6 x^{3}-x y+4 y^{2}\right) $$

Step-by-Step Solution

Answer

The subtracted polynomial is \(-5x^{3}+8xy-9y^{2}\) and its degree is 3.

Step 1: Understand subtraction of polynomials

To subtract one polynomial from another, subtract the corresponding terms in the two polynomials. That is, subtract the coefficients of the terms with the same powers.

Step 2: Subtract the polynomials

\(x^{3}+7 x y-5 y^{2} - (6 x^{3}-x y+4 y^{2}) = x^{3}+7 x y-5 y^{2} -6 x^{3}+x y-4 y^{2} = (-5x^{3}+8xy-9y^{2})\). Here, the terms \(6 x^{3}\) and \(-x y\) in the second polynomial are subtracted from their corresponding terms in the first polynomial.

Step 3: Identify the degree of resulting polynomial

The degree of a polynomial is the highest power of its variable. Here, the highest power of \(x\) in the polynomial \(-5x^{3}+8xy-9y^{2}\) is 3 which is from the term \(-5x^{3}\)

Key Concepts

Polynomial Subtraction

Subtraction of polynomials is a fundamental operation in algebra, where you subtract one polynomial from another. This process involves subtracting corresponding terms in the two polynomials. When dealing with polynomial subtraction, you perform a term-by-term subtraction, focusing on matching the like terms from each polynomial.

To start, align the polynomials and identify the terms with similar degrees. These are terms with the same variable raised to the same power. Once you've matched the like terms, subtract the coefficients of these terms.

For example, if you have the polynomials \(a_1x^n + b_1x^{n-1}\) and \(a_2x^n + b_2x^{n-1}\), the subtraction will lead to \((a_1-a_2)x^n + (b_1-b_2)x^{n-1}\). This method ensures each like term is correctly subtracted, giving the accurate result of the polynomial subtraction.

Degree of a Polynomial

The degree of a polynomial is a key characteristic, indicating the highest power of the variable in the polynomial. It's important to understand how to find this, as it affects the nature and graph of the polynomial.

In a polynomial, each term is made up of a coefficient and a variable raised to an exponent. When determining the degree, only the highest exponent on the variable is considered.

For instance, in the polynomial \(-5x^3 + 8xy - 9y^2\), the term with the highest power is \(-5x^3\). Hence, the degree of this polynomial is 3.

The degree offers insights into the number of solutions the polynomial equation can have and hints at the shape of its graph. It's a critical element in understanding polynomials better.

Coefficients

Coefficients are the numerical factors attached to the terms of a polynomial. They determine the amplitude or scale of each term in the polynomial expression.

In a general polynomial form, like \(ax^n + bx^{n-1} + cx^{n-2} + \ldots\), the numbers \(a, b, c\) are the coefficients.

When performing operations like addition, subtraction, and multiplication on polynomials, coefficients play a crucial role as you need to apply these operations directly to the coefficients of like terms.

While the coefficients may not change the degree of the polynomial, they significantly affect the graph's appearance and the function's rate of change. Understanding coefficients and their manipulation is vital for accurate polynomial operations.

Subtraction of Like Terms

Like terms in polynomials are terms that share the same variables raised to the same powers, crucial for operations like subtraction. To subtract like terms, align them based on their variables and exponents, then subtract their coefficients.

This method simplifies the polynomial and ensures only similar elements interact.

In our earlier subtraction example, matching terms like \(x^3\), \(xy\), and \(y^2\) from the polynomials \(x^3 + 7xy - 5y^2\) and \(6x^3 - xy + 4y^2\) is essential.

• For the terms \(x^3\): \(x^3 - 6x^3 = -5x^3\)
• For the terms \(xy\): \(7xy - (-xy) = 8xy\)
• For \(y^2\): \(-5y^2 - 4y^2 = -9y^2\)

By correctly subtracting like terms, your result expresses a simplified form of the polynomial. This creates a more manageable and interpretable expression.