Polynomial arithmetic: FAQ

Frequently asked questions about polynomial arithmetic

What is a polynomial?

A polynomial is a type of mathematical expression made up of one or more terms. Each term consists of a variable (usually

x

) raised to a non-negative integer exponent, and multiplied by a coefficient. For example,

3 x^{2} + 2 x - 5

is a polynomial.

Why do we need to know how to add, subtract, and multiply polynomials?

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.

How do we add or subtract two polynomials?

We can add or subtract two polynomials by combining like terms. For example, to add

3 x^{2} + 2 x - 5

and

2 x^{2} - 3 x + 1

, we combine the

x^{2}

terms, the

x

terms, and the constant terms:

(3 x^{2} + 2 x - 5) + (2 x^{2} - 3 x + 1) = 5 x^{2} - x - 4

How do we multiply a monomial by a polynomial?

To multiply a monomial (a polynomial with just one term) by a polynomial, we use the distributive property. For example, to multiply

3 x

2 x^{2} - 5 x + 6

, we multiply

3 x

by each term of the polynomial:

3 x (2 x^{2} - 5 x + 6) = 6 x^{3} - 15 x^{2} + 18 x

How do we multiply two binomials?

We can use the distributive property or an area model to multiply two binomials (polynomials with two terms). For example, to multiply

(2 x - 3) (3 x + 4)

using the distributive property we compute each product and combine the like

x

terms:

\begin{aligned} 2 x \times 3 x = 6 x^{2} \\ 2 x \times 4 = 8 x \\ - 3 \times 3 x = - 9 x \\ - 3 \times 4 = - 12 \end{aligned}

(2 x - 3) (3 x + 4) = 6 x^{2} - x - 12

What are special products of polynomials?

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:

(a + b)^{2} = a^{2} + 2 a b + b^{2}

Another common special product is the difference of two squares:

(a + b) (a - b) = a^{2} - b^{2}

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