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## Algebra 2

### Course: Algebra 2>Unit 1

Lesson 6: Special products of polynomials

# Polynomial arithmetic: FAQ

## 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, squared, plus, 2, x, minus, 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, squared, plus, 2, x, minus, 5 and 2, x, squared, minus, 3, x, plus, 1, we combine the x, squared terms, the x terms, and the constant terms:
left parenthesis, 3, x, squared, plus, 2, x, minus, 5, right parenthesis, plus, left parenthesis, 2, x, squared, minus, 3, x, plus, 1, right parenthesis, equals, 5, x, squared, minus, x, minus, 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 by 2, x, squared, minus, 5, x, plus, 6, we multiply 3, x by each term of the polynomial:
3, x, left parenthesis, 2, x, squared, minus, 5, x, plus, 6, right parenthesis, equals, 6, x, cubed, minus, 15, x, squared, plus, 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 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 terms:
\begin{aligned} &2x \times 3x = 6x^2 \\\\ &2x \times 4 = 8x \\\\ &-3 \times 3x = -9x \\\\ &-3 \times 4 = -12 \end{aligned}
So left parenthesis, 2, x, minus, 3, right parenthesis, left parenthesis, 3, x, plus, 4, right parenthesis, equals, 6, x, squared, minus, x, minus, 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:
left parenthesis, a, plus, b, right parenthesis, squared, equals, a, squared, plus, 2, a, b, plus, b, squared
Another common special product is the difference of two squares:
left parenthesis, a, plus, b, right parenthesis, left parenthesis, a, minus, b, right parenthesis, equals, a, squared, minus, b, squared