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.

A polynomial expression labeled with its parts written as: p of x is equal to a sub n times x to the nth power plus a sub n minus 1 times x to the power of n minus 1 plus ellipsis plus a sub 2 times x squared plus a sub 1 times x plus a sub 0. a sub n times x to the nth power is the leading term, a sub n is the leading coefficient and the power n is the degree. In the term a sub n minus 1 times x to the power of n inus 1, a sub n minus 1 is the coefficient. The term a sub 2 times x squared is the quadratic term, a sub 1 times x is the linear term, and a sub 0 is the constant term.

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.

Exploration

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)
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)
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.

Product of powers law

When multiplying two exponential expressions with the same base, add the exponents.

Example:

a^m \cdot a^n=a^{m+n}

a^3 \cdot a^5=a^{3+5}=a^8

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)
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.

Examples

Example 1

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

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

Worked Solution

Create a strategy

To be a polynomial, all coefficients must be constant, and all exponents must be non-negative integers.

Apply the idea

The coefficients 5,\dfrac{4}{3}, and -2 are all real number constants. The exponents on the variables are all non-negative. Therefore, this is a polynomial.

Reflect and check

There are 3 terms in this polynomial, so it is a trinomial. The leading coefficient is 5, and the degree of the polynomial is 3.

\dfrac{4}{x-3}

Worked Solution

Create a strategy

We can use the negative exponent property to rewrite this expression.

Apply the idea

\displaystyle \dfrac{4}{x-3}

\displaystyle =

\displaystyle 4(x-3)^{-1}

Negative exponent property

Because there is a variable with a negative exponent, this is not a polynomial.

Reflect and check

We will eventually learn that this is a rational expression. Rational functions have asymptotes that separate the graph into multiple pieces. Therefore, it is not continuous.

In general, whenever there is a variable in the denominator, it is not a polynomial expression or equation.

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

Worked Solution

Create a strategy

Before looking at the coefficients and exponents, we need to make sure the expression is fully simplified. For this equation, the constant can be rewritten as a fraction.

Apply the idea

\displaystyle f(x)	\displaystyle =	\displaystyle \dfrac{6x^2y}{5}-\sqrt{2}xy+0.5xy^2-4^{-1}	Given function
	\displaystyle =	\displaystyle \dfrac{6x^2y}{5}-\sqrt{2}xy+0.5xy^2-\dfrac{1}{4}	Negative exponent property

The coefficients \dfrac{6}{5}, \sqrt{2}, 0.5, and \dfrac{1}{4} are all real number constants. The exponents on the variables are all non-negative, so this is a polynomial.

Example 2

Fully simplify each polynomial expression.

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

Worked Solution

Create a strategy

To simplify the given expression, we'll start by rearranging the terms in descending order of their exponents, which makes it easier to identify like terms (terms that have the same variable raised to the same power). Then, we'll combine the coefficients of like terms.

Apply the idea

The expression \left( -5x^{3} + 7x^{2} - 4\right) + \left(3x^{3} - 9x + 2\right) is equivalent to the expression \left( -5x^{3} + 7x^{2} - 4\right) + 1\left(3x^{3} - 9x + 2\right). We can distribute the positive 1 to the second expression, and the terms will remain the same. This means we can remove all parentheses.

\displaystyle \left( -5x^{3} + 7x^{2} - 4\right) + \left(3x^{3} - 9x + 2\right)	\displaystyle =	\displaystyle -5x^{3} + 7x^{2} - 4 + 3x^{3} - 9x + 2	Distribute +1
	\displaystyle =	\displaystyle -5x^{3} + 3x^{3} + 7x^{2} -9x -4 + 2	Reorder the terms
	\displaystyle =	\displaystyle -2x^{3} + 7x^{2} - 9x - 2	Combine like terms

Therefore, the simplified expression is -2x^{3} + 7x^{2} - 9x - 2.

Reflect and check

Since the highest power of x in any term is 3, we can conclude that our simplified expression is a third-degree polynomial.

The expression has been arranged in descending order by the power of x, starting from x^{3} and proceeding to the constant term, indicating that it is in the standard form for polynomials.

Expressing polynomials in standard not only organizes the terms efficiently but also ensures that all like terms have been appropriately combined.

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

Worked Solution

Create a strategy

We can simplify this expression by combining like terms. To do so, we must be careful to apply the subtraction to each term in the second polynomial. This is the same as multiplying each term in the second expression by -1. Also, remember that like terms have the same variables with the same exponents.

Apply the idea

First, we will distribute the subtraction to the second polynomial.\begin{aligned}&\left(-7x^{3} + 5.5x^{2} - 2.1x\right) - \left(4.3x^{3} - 1.7x^{2} + 3x - 0.5\right)\\=&-7x^{3} + 5.5x^{2} - 2.1x - 4.3x^{3} + 1.7x^{2} - 3x + 0.5\end{aligned}Next, we will group like terms together.=-7x^{3}-4.3x^{3}+5.5x^{2}+1.7x^{2}-2.1x -3x +0.5Finally, we can combine like terms.= -11.3x^{3} + 7.2x^{2} - 5.1x + 0.5

Reflect and check

The different forms of the expressions highlight different things. The original form clearly identifies the two polynomials in the expression. However, the simplified form help us easily identify the leading coefficient, degree, and constant term of the entire polynomial expression.

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

Worked Solution

Create a strategy

To simplify this expression, we will apply the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial.

Apply the idea

\begin{aligned}\left(-2a^{2}+5b-3\right) \left(3a-4b\right)&=-2a^{2}\left(3a\right)-2a^{2}\left(-4b\right)+ 5b \left(3a\right)+5b\left(-4b\right)-3\left(3a\right)-3\left(-4b\right)\\&=-6a^{3} +8a^{2}b +15ab -20b^{2} -9a+12b\end{aligned}

Reflect and check

An area model visually represents the distributive property used in multiplication. Here, we used the product of powers law of exponents. This law says that when we multiply monomials, we add their exponents.

An area model with algebraic expression in each cell.

The area of each rectangle in the model represents the product of the side lengths, akin to multiplying each term of the first polynomial by each term of the second polynomial. This model confirms our simplified expression.

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

Worked Solution

Create a strategy

To simplify, we first distribute the subtraction to the last polynomial, then we can reorder and combine like terms.

Apply the idea

\begin{aligned}&\left(3y^{2} + 2x - 4\right) + \left(4x^{2} - x + 3y\right) - \left(2y - 5x + 3\right) \text{ \qquad \qquad \,\, \, \, Start with the given expression}\\= &3y^{2} + 2x - 4 + 4x^{2} - x + 3y - 2y + 5x - 3 \text{ \quad \qquad \qquad \qquad \enspace \,Distribute the subtraction}\\=&3y^{2}+4x^{2}+2x-x+5x+3y-2y-4-3 \text{ \qquad \qquad \quad \quad \quad \enspace \, Group like terms}\\=& 3y^{2} + 4x^{2} + 6x + y - 7 \text{ \qquad \qquad \qquad \quad \qquad \quad \quad \quad \qquad \qquad \, Combine like terms}\end{aligned}

The simplified expression is 3y^{2} + 4x^{2} + 6x + y - 7.

Reflect and check

It is important that we accurately distribute signs and combine like terms. To organize our work, we can underline like terms with the same color and underline the other terms with different colors.

Three lines of algebraic simplification are shown, each with step-by-step annotations for clarity. Speak to your teacher for more information.

This helps us visually represent the polynomial expression and confirm that we have simplified correctly.

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

Worked Solution

Create a strategy

To simplify this expression, we will distribute each monomial across its respective polynomial, carefully combining like terms afterward.

Apply the idea

In the given expression, 4y \left(2x^{2} - x + 5\right) + 6x \left(3y^{2} - 2y + 4\right) - 5y \left(4x - y\right), notice that 4y, 6x, and -5y are coefficients that need to be multiplied to the expressions in the parentheses adjacent to them. We can multiply these expressions individually:

4y \left(2x^{2} - x + 5\right)= 8x^{2}y - 4xy + 20y
6x \left(3y^{2} - 2y + 4\right) =18x y^{2} - 12xy + 24x
-5y \left(4x - y\right) = -20xy + 5y^{2}

Now, we can combine the results into a single polynomial and continue to simplify the expression.

\begin{aligned}&8x^{2}y - 4xy + 20y+ 18x y^{2} - 12xy + 24x -20xy + 5y^{2} \text{ \quad Rewrite as a single polynomial}\\=&8x^{2}y + 18x y^{2} + 5y^{2}- 4xy - 12xy -20xy + 24x+ 20y \text{ \quad Reorder the terms }\\=&8x^{2}y + 18x y^{2} + 5y^{2}- 36xy + 24x+ 20y \text{ \quad \qquad \qquad \qquad Combine like terms }\end{aligned}

The simplified expression is 8x^{2}y + 18x y^{2} + 5y^{2}- 36xy + 24x+ 20y.

Example 3

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

A rectangle with a length of 1.6 x plus 9 halves y and width of one half x plus 0.4 y.

Worked Solution

Create a strategy

The perimeter of a shape is the sum of its side lengths. In this case, the shape is a rectangle, so we can add the two labeled side lengths and then double the result.

Apply the idea

\displaystyle P	\displaystyle =	\displaystyle 2\left(l+w\right)	Formula for perimeter
	\displaystyle =	\displaystyle 2\left(1.6x + \dfrac{9}{2}y + \dfrac{1}{2}x + 0.4y\right)	Substitute expressions for the length and width
	\displaystyle =	\displaystyle 2\left(1.6x + 4.5y + 0.5x + 0.4y\right)	Rewrite the fractions as decimals
	\displaystyle =	\displaystyle 2\left(2.1x + 4.9y\right)	Combine like terms
	\displaystyle =	\displaystyle 4.2x + 9.8y	Distributive property

Reflect and check

There are benefits to having the rectangle's perimeter in both its original and simplified forms. Although these forms mean the same thing, using the simplified form makes it easier to substitute values for x and y. On the other hand, the original form helps us easily see the dimensions of the rectangle.

Example 4

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}.

A rectangular swimming pool with length 16 yards, and width 6 yards. A path forms a rectangle around the pool x yards in width from each side of the pool.

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

Worked Solution

Create a strategy

The area of the path, A_{\text{path}}, will be the area of the larger rectangle minus the area of the pool.

The area of pool, A_{\text{pool}}, is represented by the inner green rectangle which has a length of 16\text{ yds} and a width of 6\text{ yds}.

The area of the large rectangle, A_{\text{large rectangle}}, is a combination of the area of the path and the area of the pool. We can see that the large rectangle has a length of 16\text{ yds} plus an x\text{ yds} on either side of the pool. This gives us a length of x+16+x = \left(16+2x\right)\text{ yds} We can apply the same logic to the width of the large rectangle to find a width of \left(6+2x\right)\text{ yds}.

Apply the idea

\displaystyle A_{\text{path}}	\displaystyle =	\displaystyle A_{\text{large rectangle}}-A_{\text{pool}}	Equation for area of the path
	\displaystyle =	\displaystyle \left(16+2x\right)\left(6+2x\right)-\left(16\right)\left(6\right)	Substitute expressions for each area
	\displaystyle =	\displaystyle 96 +32x+12x+4x^2-96	Distributive property
	\displaystyle =	\displaystyle 4x^2+44x	Combine like terms

A_{\text{path}}=\left(4 x^{2} + 44 x\right)\text{ yd}^2

Reflect and check

There are good things about knowing the rectangular path's area in both its original unsimplified form, and simplified forms.

The original form of the rectangular path's area, A_{\text{path}}=\left(16+2x\right)\left(6+2x\right)-\left(16\right)\left(6\right), reveals several important details. It shows the dimensions of both the path and the pool, the combined area of the pool and the path, and the area of the pool alone.

In contrast, the simplified version is specifically useful for calculating the area of the path alone.

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.

Outcomes

A2.EO.3

The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

A2.EO.3a

Determine sums, differences, and products of polynomials in one and two variables.

A2.EO.3d

Represent and demonstrate equality of polynomial expressions written in different forms and verify polynomial identities including the difference of squares, sum and difference of cubes, and perfect square trinomials.

3.01 Operations with polynomials

Ideas

Operations with polynomials

Exploration

Examples

Example 1

Example 2

Example 3

Example 4

Outcomes

A2.EO.3

A2.EO.3a

A2.EO.3d

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