Question

Extend the polynomial system to include subtraction of polynomials. (Hint: You may find it helpful to define a generic negation operation.)

Short answer

Subtracting polynomials involves adding the negation of the subtracted polynomial.

Step-by-step solution

Understanding Polynomial Subtraction

To subtract polynomials, recognize it as adding the negation of a polynomial. For example, to subtract polynomial \( Q(x) \) from polynomial \( P(x) \), compute \( P(x) + (-Q(x)) \).

Defining the Negation Operation

The negation of a polynomial changes the sign of each term in the polynomial. If \( Q(x) = q_nx^n + q_{n-1}x^{n-1} + ... + q_1x + q_0 \), then \(-Q(x) = -q_nx^n - q_{n-1}x^{n-1} - ... - q_1x - q_0 \).

Applying Negation to the Polynomial

Given a polynomial \( Q(x) = 3x^2 + 2x - 5 \), its negation is \(-Q(x) = -3x^2 - 2x + 5 \).

Performing the Subtraction

To subtract \( Q(x) \) from \( P(x) \), compute \( P(x) + (-Q(x)) \). If \( P(x) = 5x^2 + x + 4 \), then \( P(x) - Q(x) = (5x^2 + x + 4) + (-3x^2 - 2x + 5) = 2x^2 - x + 9 \).

Key concepts

Negation Operation

The negation operation is a crucial step in polynomial subtraction. When you negate a polynomial, you change the sign of each of its terms. This means converting positive terms into negative and vice versa. This operation allows us to transform a subtraction problem into an addition problem, which can be easier to handle.
For instance, consider the polynomial \( Q(x) = 3x^2 + 2x - 5 \). The negation of this polynomial is achieved by changing the sign of each term, resulting in \(-Q(x) = -3x^2 - 2x + 5 \). Notice how each term in \( Q(x) \) has had its sign flipped to produce \(-Q(x) \). By handling subtraction in this manner, we can use the familiar rules of polynomial addition instead.

Polynomial Addition

Once you have negated one of the polynomials in a subtraction problem, the task becomes a matter of polynomial addition. Adding polynomials involves combining like terms, which are terms that have the same variables raised to the same power. The coefficients of these terms are added together.
For example, if we are to subtract \( Q(x) = 3x^2 + 2x - 5 \) from \( P(x) = 5x^2 + x + 4 \), we rewrite it as an addition problem: \[ P(x) - Q(x) = P(x) + (-Q(x)) \]. After negating \( Q(x) \) as \(-3x^2 - 2x + 5 \), add them up:

• Combine the \( x^2 \) terms: \( 5x^2 - 3x^2 = 2x^2 \)
• Combine the \( x \) terms: \( x - 2x = -x \)
• Combine the constant terms: \( 4 + 5 = 9 \)

This results in \( P(x) - Q(x) = 2x^2 - x + 9 \). Therefore, what seemed challenging initially is simplified into straightforward arithmetic.

Algebraic Expressions

Algebraic expressions like polynomials are mathematical sentences that include constants, variables, and exponents. Understanding these components is essential when performing operations such as addition and subtraction. Each term in an algebraic expression consists of a coefficient (which is a constant or a number), a variable, and an exponent.
For example, in the algebraic expression \( 3x^2 + 2x - 5 \), the terms are \( 3x^2 \), \( 2x \), and \(-5 \). Here, \( 3 \), \( 2 \), and \(-5 \) are coefficients; \( x \) is the variable; and \( 2 \), \( 1 \) (implied for \( 2x \)), and \( 0 \) (since \(-5 \) is a constant term without a variable) are the exponents.
When performing operations on algebraic expressions, like subtraction by negation and then addition, it's important to identify and correctly manipulate these terms. This kind of clarity ensures accuracy in simplifying or rearranging these expressions for any mathematical purpose.