Polynomial arithmetic: FAQ

A polynomial is a type of mathematical expression made up of one or more terms. Each term consists of a variable (usually $x$x) raised to a non-negative integer exponent, and multiplied by a coefficient. For example, $3x^2 + 2x - 5$3, x, squared, plus, 2, x, minus, 5 is a polynomial.

Polynomial arithmetic is important for solving a variety of problems in mathematics, physics, engineering, and more. For example, knowing how to multiply polynomials can help us factor them, which in turn can be useful for solving polynomial equations.

We can add or subtract two polynomials by combining like terms. For example, to add $3x^2 + 2x - 5$3, x, squared, plus, 2, x, minus, 5 and $2x^2 - 3x + 1$2, x, squared, minus, 3, x, plus, 1, we combine the $x^2$x, squared terms, the $x$x terms, and the constant terms:

To multiply a monomial (a polynomial with just one term) by a polynomial, we use the distributive property. For example, to multiply $3x$3, x by $2x^2 - 5x + 6$2, x, squared, minus, 5, x, plus, 6, we multiply $3x$3, x by each term of the polynomial:

We can use the distributive property or an area model to multiply two binomials (polynomials with two terms). For example, to multiply $(2x - 3)(3x + 4)$left parenthesis, 2, x, minus, 3, right parenthesis, left parenthesis, 3, x, plus, 4, right parenthesis using the distributive property we compute each product and combine the like $x$x terms:

So $(2x - 3)(3x + 4) = 6x^2 - x - 12$left parenthesis, 2, x, minus, 3, right parenthesis, left parenthesis, 3, x, plus, 4, right parenthesis, equals, 6, x, squared, minus, x, minus, 12.

There are certain polynomial products that occur frequently in mathematics, and it's helpful to recognize them.

For example, the square of a binomial is:

Another common special product is the difference of two squares: