In Algebra 1, we learned the following definitions related to polynomials:

The term which has a fixed value and no variables is called the **constant term**. The term with highest exponent on the variable is called the **leading term**, and the exponent of the leading term is the **degree** of the polynomial.

The standard form of a polynomial is a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_1 x + a_0, where n is a non-negative integer and each a_i is a coefficient.

Polynomials can also have names specific to the number of terms they have. A **monomial** is a polynomial with one term. A **binomial** is a polynomial with two terms. A **trinomial** is a polynomial with three terms.

They can also be named based on their degree. A degree 0 polynomial is **constant**. A degree 1 polynomial is **linear**. A degree 2 polynomial is **quadratic**. 3 is **cubic**, 4 is **quartic** and so on.

In the definition of polynomials, the coefficients are multiplied to the variables, the variables are raised to non-negative integer powers, and the terms are added and subtracted together. This allows the function to be one, smooth curve with no breaks, holes, sharp turns, or stopping points.

Use technology to graph the following functions:

- y=x^2+2x-1
- y=-3x^{5}-4x^{4}+2x^{3}+x^{2}-0.2x
- y=4x^3+\frac{1}{x}
- y=2\sqrt{x}
- y=|x|

- Determine if the function is a polynomial and explain your answer.

One way to determine if an expression or equation is a polynomial is to examine its graph. The graph of a polynomial is a function with one smooth curve over a **continuous domain**. A polynomial will not have a negative exponent on a variable, a rational exponent on a variable, or have a variable in absolute value bars.

Polynomials are **closed** under addition, subtraction, and multiplication. This means that the sum, product, or difference of polynomials will also be a polynomial.

When adding and subtracting polynomials, we use the method of combining like terms.

Addition: if we assume m<n,

General example | \left(a_{n}x^{n}+\ldots +a_{0}\right)+\left(b_{m}x^{m}+\ldots +b_{0}\right)=\\a_{n}x^n+\ldots +\left(a_m+b_m\right)x^m+\ldots +\left(a_0+b_0\right) |
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Numerical example | \left(2.7x^5-1.8x^3+0.9x-2\right)+\left(3.8x^4+2x^3-x+5.1\right)=\\ 2.7x^5+3.8x^4+0.2x^3-0.1x+3.1 |

By definition, m and n will be non-negative integers, and the coefficients will remain constant. Therefore, the result is another polynomial.

Subtraction will work the same way as addition. Assuming m<n,

General example | \left(a_{n}x^{n}+\ldots+a_{0}\right)-\left(b_{m}x^{m}+\ldots +b_{0}\right)=\\a_{n}x^n+\ldots +\left(a_m-b_m\right)x^m+\ldots +\left(a_0-b_0\right) |
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Numerical example | \left(\frac{2}{3}x^4+\frac{3}{5}x^2+\frac{1}{2}\right)-\left(\frac{1}{4}x^3+\frac{3}{5}x^2+\frac{4}{7}x+\frac{3}{4}\right)=\\ \frac{2}{3}x^4-\frac{1}{4}x^3-\frac{4}{7}x-\frac{1}{4} |

The coefficients will remain constant, and the exponents will be non-negative integers. The result is another polynomial.

To multiply polynomials, we apply the distributive property which will require us to use the **product of powers law** of exponents when multiplying variables.

General example | \left(a_{n}x^{n}+\ldots+a_{0}\right)\left(b_{m}x^{m}+\ldots +b_{0}\right)=\\ \left(a_{n}b_{m}\right)x^{n+m}+\ldots+\left(a_{n}b_0\right)x^n+\ldots+\left(a_{0}b_m\right)x^m+\ldots+\left(a_0b_0\right) |
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Numerical example | \left(5x^2+2x+1\right)\left(x^2-3x+5\right)= 5x^4-13x^3+20x^2+7x+5 |

Because n and m were non-negative integers, n+m will also be non-negative. The exponents will still be constants, so the result is another polynomial.

Determine whether each of the following can be classified as a polynomial.

a

y=5x^2y+\dfrac{4}{3}xy-2y

Worked Solution

b

\dfrac{4}{x-3}

Worked Solution

c

f(x)=\dfrac{6x^2y}{5}-\sqrt{2}xy+0.5xy^2-4^{-1}

Worked Solution

Fully simplify each polynomial expression.

a

\left( -5x^{3} + 7x^{2} - 4\right) + \left(3x^{3} - 9x + 2\right)

Worked Solution

b

\left(-7x^{3} + 5.5x^{2} - 2.1x\right) - \left(4.3x^{3} - 1.7x^{2} + 3x - 0.5\right)

Worked Solution

c

\left(-2a^{2}+5b-3\right) \left(3a-4b\right)

Worked Solution

d

\left(3y^{2} + 2x - 4\right) + \left(4x^{2} - x + 3y\right) - \left(2y - 5x + 3\right)

Worked Solution

e

4y \left(2x^{2} - x + 5\right) + 6x \left(3y^{2} - 2y + 4\right) - 5y \left(4x - y\right)

Worked Solution

Form a fully simplified polynomial expression for the perimeter of the rectangle shown.

Worked Solution

A rectangular swimming pool is 16\text{ yds} long and 6\text{ yds} wide. It is surrounded by a pebble path of uniform width x\text{ yds}.

Find an expression for the area of the path in terms of x. Fully simplify your answer.

Worked Solution

Idea summary

A polynomial is the sum or difference of terms which have variables raised to non-negative integer powers and coefficients that are constant.

A polynomial written in different forms can make some things easier or harder to see. To determine if two polynomials written in different forms are equivalent, we can use visual aids like area models or algebra tiles.