Compare\begin{aligned} & & 2x^3 & &+&4x^2 & + & 0x & +5 \\ + & & 0x^3& &+ & 3x^2& + & 2x & +3 \\ \hline \\ & & 2x^3 & & + & 7x^2 & + & 2x & +8 \end{aligned}to\begin{aligned} & & 2 && 4 && 0 && 5 \\ + & & && 3 && 2 && 3 \\ \hline \\ & & 2 && 7 && 2 && 8 \end{aligned}

Create an addition problem like the example provided where the sum of the coefficients is greater than 9. What happens?
Create and solve a subtraction problem using the vertical algorithm. Do polynomials behave the same as numbers when subtracting?

We can use algebra tiles to model sums and differences of polynomials.

The difference (6x^2 + 4x - 5) - (4x^2 - 2x +3) can be modeled with algebra tiles. Lining up like terms, vertically, we can write:

\begin{aligned} & &6x^2 &+4x &-5 \\ - & &4x^2 &-2x &+3 \\ \hline \\ \end{aligned}

The subtraction can be viewed as the expression:

(6x^2 + 4x - 5) + (-1)(4x^2 - 2x +3)

The image show algebra tile. Six +x squared tiles, four +x tiles and five -1 tiles minus four +x squared tiles, two -x tiles, and three +1 tiles.

Using the opposites of the expression 4x^2 - 2x +3 with the algebra tiles, we get the expression:

(6x^2 + 4x - 5) + (-4x^2 + 2x - 3)

Equivalently, distributing the -1:

\begin{aligned} & & 6x^2 &+4x &-5 \\ + & &-4x^2 &+2x &-3 \\ \hline \\ & & 2x^2 &+6x &-8 \end{aligned}

Creating zero pairs and combining like terms with the algebra tiles, we are left with the expression:

2x^2 + 6x - 8

Therefore, the difference between the revenue from the gaming computers can be modeled by 2x^2 + 6x - 8.

The image show algebra tile. Six +x squared tiles, four +x tiles and five -1 tiles plus four -x squared tiles, two +x tiles, and three -1 tiles equals two +x squared tiles, six +x tiles, and eight -1 tiles.

Adding and subtracting polynomials creates more polynomials. The following vocabulary is helpful to know when working with polynomials:

Standard form (of a polynomial)

A way of writing a polynomial expression; 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.

Degree of a polynomial

The largest exponent or the largest sum of exponents of a term within a polynomial

Leading term

The term in a polynomial with the highest exponent of the variable

Leading coefficient

The coefficient of the first term of a polynomial written in descending order of exponents

Monomial

A polynomial with one term

Binomial

A polynomial with two terms

Trinomial

A polynomial with three terms

Examples

Example 1

Consider the polynomial

3x - 6 + x^{2}

Rewrite the expression in standard form.

Worked Solution

Create a strategy

Recall that the standard form of a polynomial is written with the terms in order from the term with the highest variable exponent to the lowest. We can use the commutative property to change the order of the polynomials.

Apply the idea

x^{2} + 3x - 6

State the degree of the polynomial.

Worked Solution

Apply the idea

x^{2} + 3x - 6 is a polynomial of degree 2.

Reflect and check

Since the polynomial has 3 terms, it may be called a trinomial.

Identify the quadratic term, the linear term, and the constant term of the polynomial.

Worked Solution

Apply the idea

Quadratic term: x^{2}
Linear term: 3x
Constant term: -6

Reflect and check

Polynomials of degree 2 are called quadratic polynomials.

Example 2

Consider the polynomials x^3-6x+2 and x^2+9x+7.

Find the sum of the two polynomials.

Worked Solution

Create a strategy

Since we want to find the sum of the two polynomials, combine the like terms.

Apply the idea

\displaystyle \text{Sum}	\displaystyle =	\displaystyle \left(x^3-6x+2\right)+\left(x^2+9x+7\right)	Add the polynomials together
	\displaystyle =	\displaystyle x^3-6x+2+x^2+9x+7	Remove the parentheses (Associative Property)
	\displaystyle =	\displaystyle x^3+x^2+3x+9	Combine the like terms

Reflect and check

If we want to use the vertical algorithm method, we need to make sure we correctly align the like terms.

\begin{aligned} & & x^3 & & & & - & 6x & + 2 \\ + & & & & & x^2 & + & 9x & + 7 \\ \hline \\ & & x^3 & & + & x^2 & + & 3x & + 9 \end{aligned}

Explain why the sum of two polynomials is also a polynomial.

Worked Solution

Apply the idea

By definition, a polynomial is the sum or difference of terms which have variables raised to non-negative integer powers and which have coefficients that may be real or complex.

Adding one polynomial to the other is the same as adding more terms to one polynomial. This doesn't change the fact that it is a polynomial, so the sum of two polynomials will always be a polynomial.

Reflect and check

We can use the same explanation for why the difference between two polynomials is also a polynomial, and we can extend this explanation to include the sum or difference of any number of polynomials.

Example 3

Simplify the expression:\left(3x^2-5x+1\right)-\left(x^2+7x-10\right)

Worked Solution

Apply the idea

\displaystyle \left(3x^2-5x+1\right)-\left(x^2+7x-10\right)	\displaystyle =	\displaystyle 3x^2-5x+1-x^2-7x+10	Distribute the subtraction
	\displaystyle =	\displaystyle \left(3x^2-x^2\right)+\left(-5x-7x\right)+\left(1+10\right)	Group the like terms together
	\displaystyle =	\displaystyle 2x^2-12x+11	Simplify

Reflect and check

A color-coded visualization helps confirm that our distribution is correct.

A color coded visualization showing how to simplify the expression left parenthesis 3 x squared minus 5 x plus 1 right parenthesis minus left parenthesis x squared plus 7 x minus 10 right parenthesis. Speak to your teacher for more information.

Example 4

Write at least two equivalent expressions for the length of a fence around a rectangular yard with a length of 2x^{3} + 5 feet and a width of x^{2} + 6 feet.

Worked Solution

Create a strategy

Draw a diagram of the yard and label its length and width. Use the length and width to write the perimeter.

A rectangle with a width of x squared plus 6 feet and a length of 2 x cubed plus 5 feet.

Apply the idea

The perimeter of the yard is the sum of the sides. One expression for the length of the fence would be

(2x^{3} + 5) + (x^{2} + 6) +(2x^{3} + 5) + (x^{2} + 6) \text{ ft}

By finding the sum of the binomials representing the length and width of the yard, another expression would be

2x^{2}+22+4x^{3} \text{ ft}

The expression written in standard form begins with the term with the highest degree and ends with the term with the lowest degree, shown below:

4x^{3} + 2x^{2} + 22 \text{ ft}

Example 5

Model and simplify (8 x^{2} + 4) + (12 x^{2} + 6x + 6).

Worked Solution

Create a strategy

Represent each term with tiles and combine the similar terms.

Apply the idea

The expression 8x^{2} + 4 is the same as having eight x^{2} tiles and four unit (or 1) tiles.

The image shows eight x squared and four unit tiles.

The expression 12x^{2} + 6x + 6 is the same as having twelve x^{2} tiles, six x tiles, and 6 unit (or 1) tiles.

The image shows twelve x squared, six x, and six unit tiles.

The expression (8 x^{2} + 4) + (12 x^{2} + 6x + 6) can be represented as:

The image shows eight x squared and four unit tiles add twelve x squared, six x, and six unit tiles.

Combine the like terms and count each type of tile.

The image shows twenty x squared, six x, and ten unit tiles.

So, there are twenty x^{2}, six x, and ten unit tiles.

The result is 20 x^{2} + 6 x + 10.

Idea summary

We add polynomials by combining like terms. We subtract polynomials by adding the negative terms.

Outcomes

A.EO.2

The student will perform operations on and factor polynomial expressions in one variable.

A.EO.2a

Determine sums and differences of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models.

6.01 Add and subtract polynomials

Ideas

Add and subtract polynomials

Exploration

Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Outcomes

A.EO.2

A.EO.2a

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