Polynomial Operations and Simplifications

Front

What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, 3x^2 + 2x - 5 is a polynomial.

Front

What are like terms in a polynomial?

Back

Like terms are terms that have the same variable raised to the same power. For example, in the polynomial 4x^2 + 3x - 2x^2, the terms 4x^2 and -2x^2 are like terms.

Front

How do you add polynomials?

Back

To add polynomials, combine like terms by adding their coefficients. For example, (3x^2 + 2x) + (4x^2 + 5) = (3x^2 + 4x^2) + 2x + 5 = 7x^2 + 2x + 5.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial 5x^3 + 2x^2 - x + 7, the degree is 3.

Front

What is the difference between a monomial and a polynomial?

Back

A monomial is a polynomial with only one term, while a polynomial can have multiple terms. For example, 3x is a monomial, and 3x + 2 is a polynomial.

Front

How do you subtract polynomials?

Back

To subtract polynomials, distribute the negative sign to the second polynomial and then combine like terms. For example, (3x^2 + 2x) - (x^2 + 4) = (3x^2 - x^2) + 2x - 4 = 2x^2 + 2x - 4.

Front

What is the product of two binomials?

Back

The product of two binomials can be found using the distributive property or the FOIL method (First, Outside, Inside, Last). For example, (a + b)(c + d) = ac + ad + bc + bd.

Front

What is the result of multiplying a polynomial by a monomial?

Back

To multiply a polynomial by a monomial, distribute the monomial to each term of the polynomial. For example, 3x(2x^2 + x - 4) = 6x^3 + 3x^2 - 12x.

Front

What is the process of simplifying a polynomial?

Back

Simplifying a polynomial involves combining like terms and reducing the expression to its simplest form. For example, 2x^2 + 3x - 4 + x^2 - 2x simplifies to 3x^2 + x - 4.

Front

What is the significance of the leading coefficient in a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It affects the end behavior of the polynomial's graph.