key term - Polynomial Arithmetic

Definition

Polynomial arithmetic refers to the fundamental operations performed on polynomial expressions, including addition, subtraction, multiplication, and division. These operations are essential for manipulating and simplifying polynomial functions, which are a crucial component of power functions and polynomial functions.

5 Must Know Facts For Your Next Test

1. Polynomial addition involves combining like terms by adding their coefficients.
2. Polynomial subtraction is performed by changing the sign of the subtrahend and then adding the polynomials.
3. Polynomial multiplication is carried out by multiplying each term of one polynomial with each term of the other polynomial.
4. Polynomial division can be performed using the long division algorithm or the synthetic division method.
5. The degree of a polynomial is the highest power of the variable in the polynomial expression.

Review Questions

• Explain the process of adding two polynomial expressions.
  • To add two polynomial expressions, you need to combine the like terms by adding their coefficients. This means that for each variable raised to the same power, you add the coefficients of those terms. For example, to add $3x^2 + 2x - 5$ and $4x^2 - x + 7$, you would combine the $x^2$ terms by adding their coefficients (3 + 4 = 7), the $x$ terms by adding their coefficients (2 + (-1) = 1), and the constant terms by adding -5 and 7 to get 2. The resulting sum would be $7x^2 + x + 2$.
• Describe the relationship between polynomial arithmetic and power functions.
  • Power functions, which are functions of the form $f(x) = x^n$, where $n$ is a real number, are a specific type of polynomial function. Polynomial arithmetic is crucial for manipulating and simplifying power functions, as the operations of addition, subtraction, multiplication, and division are used to perform various transformations on these functions. For example, the product of two power functions with the same base is another power function with the exponents added, and the quotient of two power functions with the same base is another power function with the exponents subtracted.
• Analyze how polynomial arithmetic is used to solve problems involving polynomial functions.
  • Polynomial arithmetic is essential for solving problems involving polynomial functions, as it allows for the manipulation and simplification of these functions. For instance, when graphing a polynomial function, you may need to factor the polynomial or find its zeros, both of which require the use of polynomial arithmetic operations. Additionally, when solving polynomial equations, you often need to perform polynomial division to isolate the variable, which relies on the principles of polynomial arithmetic. Furthermore, polynomial arithmetic is used to perform transformations on polynomial functions, such as shifting, stretching, or reflecting the graph, which is crucial for understanding the behavior of these functions.

Related terms

Polynomial:

A polynomial is an algebraic expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers.

Power Function:

A power function is a function of the form $f(x) = x^n$, where $n$ is a real number.

Polynomial Function:

A polynomial function is a function that can be expressed as the sum of a finite number of terms, each of which is the product of a constant and a variable raised to a non-negative integer power.