Chapter Notes: Polynomial Arithmetic

Table of Contents
1. Understanding Polynomial Structure
2. Adding Polynomials
3. Subtracting Polynomials
4. Multiplying Polynomials
5. Special Products
6. Dividing Polynomials
7. Applications of Polynomial Arithmetic
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Polynomials are algebraic expressions that combine variables and constants using addition, subtraction, and multiplication. Just like you can add, subtract, multiply, and divide numbers, you can perform arithmetic operations on polynomials. Understanding polynomial arithmetic forms the foundation for solving complex equations, graphing functions, and working with formulas in science and engineering. In this chapter, you will learn how to add, subtract, multiply, and divide polynomials using systematic methods that build on the algebraic skills you already know.

Understanding Polynomial Structure

Before performing operations on polynomials, it's important to understand their structure. A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where the exponents on variables are whole numbers.

Each part of a polynomial separated by a plus or minus sign is called a term. For example, in the polynomial \( 3x^2 + 5x - 7 \), there are three terms: \( 3x^2 \), \( 5x \), and \( -7 \).

Within each term:

• The coefficient is the numerical factor (like the 3 in \( 3x^2 \))
• The variable is the letter representing an unknown quantity (like \( x \))
• The exponent or power tells how many times the variable is multiplied by itself (like the 2 in \( x^2 \))
• The degree of a term is the exponent on the variable

The degree of a polynomial is the highest degree among all its terms. For instance, \( 3x^2 + 5x - 7 \) has degree 2 because the highest exponent is 2.

Polynomials are typically written in standard form, meaning the terms are arranged from highest degree to lowest degree. For example, \( 4x^3 - 2x^2 + 7x + 1 \) is in standard form.

Like terms are terms that have exactly the same variable raised to the same power. Only the coefficients can differ. For example, \( 5x^2 \) and \( -3x^2 \) are like terms, but \( 5x^2 \) and \( 5x \) are not like terms because the exponents differ.

Adding Polynomials

Adding polynomials involves combining like terms. The process is similar to combining like terms in simpler algebraic expressions, but you must pay careful attention to matching exponents.

The key principle: you can only add coefficients of like terms. Terms with different exponents must remain separate.

Vertical Method for Adding Polynomials

One approach is to align the polynomials vertically, placing like terms in the same column, then add the coefficients in each column.

Example:  Add the polynomials \( (3x^2 + 5x - 4) \) and \( (2x^2 - 3x + 7) \).

What is the sum?

Solution:

Align like terms vertically:

\( \phantom{+} 3x^2 + 5x - 4 \)
\( + 2x^2 - 3x + 7 \)

Add the coefficients in each column:

For \( x^2 \) terms: \( 3 + 2 = 5 \), giving \( 5x^2 \)
For \( x \) terms: \( 5 + (-3) = 2 \), giving \( 2x \)
For constant terms: \( -4 + 7 = 3 \), giving \( 3 \)

The sum is \( 5x^2 + 2x + 3 \).

The final answer is \( 5x^2 + 2x + 3 \).

Horizontal Method for Adding Polynomials

Another approach is to remove parentheses, group like terms, and then combine them.

Example:  Add \( (4x^3 - 2x + 5) \) and \( (-x^3 + 3x^2 + 2x - 1) \).

What is the sum?

Solution:

Write the expression without changing any signs:
\( 4x^3 - 2x + 5 - x^3 + 3x^2 + 2x - 1 \)

Group like terms together:
\( (4x^3 - x^3) + 3x^2 + (-2x + 2x) + (5 - 1) \)

Combine each group:
\( 3x^3 + 3x^2 + 0x + 4 \)

Simplify by removing the zero term:
\( 3x^3 + 3x^2 + 4 \)

The sum is \( 3x^3 + 3x^2 + 4 \).

Subtracting Polynomials

Subtracting polynomials is similar to adding them, but requires an important extra step: you must distribute the negative sign (or multiply by -1) to every term in the polynomial being subtracted. This changes the sign of each term.

Remember: subtracting a polynomial means adding its opposite. The opposite of a polynomial is formed by changing the sign of every term.

Vertical Method for Subtracting Polynomials

Example:  Subtract \( (2x^2 - 5x + 3) \) from \( (7x^2 + 2x - 1) \).

What is the difference?

Solution:

Align the polynomials vertically:

\( \phantom{-} 7x^2 + 2x - 1 \)
\( - (2x^2 - 5x + 3) \)

Change the sign of each term in the second polynomial:
\( \phantom{-} 7x^2 + 2x - 1 \)
\( - 2x^2 + 5x - 3 \)

Add the coefficients in each column:
For \( x^2 \) terms: \( 7 - 2 = 5 \), giving \( 5x^2 \)
For \( x \) terms: \( 2 + 5 = 7 \), giving \( 7x \)
For constant terms: \( -1 - 3 = -4 \), giving \( -4 \)

The difference is \( 5x^2 + 7x - 4 \).

Horizontal Method for Subtracting Polynomials

Example:  Subtract \( (3x^3 + 2x - 7) \) from \( (5x^3 - x^2 + 4) \).

What is the result?

Solution:

Write the subtraction problem:
\( (5x^3 - x^2 + 4) - (3x^3 + 2x - 7) \)

Distribute the negative sign to each term in the second polynomial:
\( 5x^3 - x^2 + 4 - 3x^3 - 2x + 7 \)

Group like terms:
\( (5x^3 - 3x^3) - x^2 - 2x + (4 + 7) \)

Combine like terms:
\( 2x^3 - x^2 - 2x + 11 \)

The difference is \( 2x^3 - x^2 - 2x + 11 \).

Multiplying Polynomials

Multiplying polynomials requires applying the distributive property repeatedly. Every term in the first polynomial must be multiplied by every term in the second polynomial, then like terms are combined.

Multiplying a Monomial by a Polynomial

A monomial is a polynomial with only one term. When multiplying a monomial by a polynomial, distribute the monomial to each term of the polynomial.

Example:  Multiply \( 3x \) by \( (4x^2 - 5x + 2) \).

What is the product?

Solution:

Distribute \( 3x \) to each term inside the parentheses:

\( 3x \cdot 4x^2 = 12x^3 \)
\( 3x \cdot (-5x) = -15x^2 \)
\( 3x \cdot 2 = 6x \)

Combine all the products:
\( 12x^3 - 15x^2 + 6x \)

The product is \( 12x^3 - 15x^2 + 6x \).

Multiplying Two Binomials

A binomial is a polynomial with exactly two terms. The most common method for multiplying two binomials is the FOIL method, which stands for First, Outer, Inner, Last-referring to which terms you multiply together.

• First: multiply the first terms of each binomial
• Outer: multiply the outer terms
• Inner: multiply the inner terms
• Last: multiply the last terms of each binomial

Example:  Multiply \( (2x + 3) \) by \( (x - 5) \).

What is the product?

Solution:

Apply the FOIL method:

First: \( 2x \cdot x = 2x^2 \)
Outer: \( 2x \cdot (-5) = -10x \)
Inner: \( 3 \cdot x = 3x \)
Last: \( 3 \cdot (-5) = -15 \)

Combine all four products:
\( 2x^2 - 10x + 3x - 15 \)

Combine like terms:
\( 2x^2 - 7x - 15 \)

The product is \( 2x^2 - 7x - 15 \).

Multiplying Larger Polynomials

When multiplying polynomials with more than two terms, you must distribute every term in the first polynomial to every term in the second polynomial. This is sometimes called the distributive method or horizontal method.

Example:  Multiply \( (x + 2) \) by \( (x^2 - 3x + 4) \).

What is the product?

Solution:

Distribute each term of \( (x + 2) \) to every term of \( (x^2 - 3x + 4) \):

First, distribute \( x \):
\( x \cdot x^2 = x^3 \)
\( x \cdot (-3x) = -3x^2 \)
\( x \cdot 4 = 4x \)

Next, distribute \( 2 \):
\( 2 \cdot x^2 = 2x^2 \)
\( 2 \cdot (-3x) = -6x \)
\( 2 \cdot 4 = 8 \)

Write all products together:
\( x^3 - 3x^2 + 4x + 2x^2 - 6x + 8 \)

Combine like terms:
\( x^3 + (-3x^2 + 2x^2) + (4x - 6x) + 8 \)
\( x^3 - x^2 - 2x + 8 \)

The product is \( x^3 - x^2 - 2x + 8 \).

Vertical Method for Multiplying Polynomials

Another effective technique, especially for larger polynomials, is the vertical method. This approach is similar to how you multiply multi-digit numbers.

Example:  Multiply \( (2x^2 - 3x + 1) \) by \( (x + 4) \) using the vertical method.

What is the product?

Solution:

Arrange the polynomials vertically:

\( \phantom{×(x+4)} 2x^2 - 3x + 1 \)
\( \phantom{2x^2 - 3} × \phantom{.} (x + 4) \)

Multiply each term of the top polynomial by 4:
\( 8x^2 - 12x + 4 \)

Multiply each term of the top polynomial by \( x \) (shift one position left):
\( 2x^3 - 3x^2 + x \)

Add the two results vertically, aligning like terms:
\( \phantom{+} 2x^3 - 3x^2 + \phantom{0}x + 0 \)
\( \phantom{2x^3+} 8x^2 - 12x + 4 \)

Combine:
\( 2x^3 + 5x^2 - 11x + 4 \)

The product is \( 2x^3 + 5x^2 - 11x + 4 \).

Special Products

Certain polynomial multiplication patterns occur so frequently that recognizing them can save time and reduce errors. These are called special products.

Square of a Binomial

When you square a binomial (multiply it by itself), a specific pattern emerges:

In words: the square of a binomial equals the square of the first term, plus (or minus) twice the product of both terms, plus the square of the second term.

Example:  Expand \( (3x + 5)^2 \).

What is the expanded form?

Solution:

Use the pattern \( (a + b)^2 = a^2 + 2ab + b^2 \) where \( a = 3x \) and \( b = 5 \):

First term squared: \( (3x)^2 = 9x^2 \)
Twice the product: \( 2(3x)(5) = 30x \)
Second term squared: \( 5^2 = 25 \)

Combine all parts:
\( 9x^2 + 30x + 25 \)

The expanded form is \( 9x^2 + 30x + 25 \).

Difference of Squares

When you multiply the sum and difference of the same two terms, you get the difference of squares:

Notice that the middle terms cancel out, leaving only the difference between the squares of the two terms.

Example:  Multiply \( (4x + 7)(4x - 7) \).

What is the product?

Solution:

Use the pattern \( (a + b)(a - b) = a^2 - b^2 \) where \( a = 4x \) and \( b = 7 \):

First term squared: \( (4x)^2 = 16x^2 \)
Second term squared: \( 7^2 = 49 \)

Apply the difference of squares formula:
\( 16x^2 - 49 \)

The product is \( 16x^2 - 49 \).

Dividing Polynomials

Division of polynomials can be performed in several ways depending on the complexity of the problem. The two main methods are dividing by a monomial and long division.

Dividing a Polynomial by a Monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial separately.

Example:  Divide \( 12x^3 - 18x^2 + 6x \) by \( 3x \).

What is the quotient?

Solution:

Divide each term by \( 3x \):

\( \frac{12x^3}{3x} = 4x^2 \)
\( \frac{-18x^2}{3x} = -6x \)
\( \frac{6x}{3x} = 2 \)

Combine the results:
\( 4x^2 - 6x + 2 \)

The quotient is \( 4x^2 - 6x + 2 \).

Polynomial Long Division

When dividing a polynomial by another polynomial with two or more terms, use polynomial long division, which works similarly to numerical long division.

The steps for polynomial long division are:

1. Divide the leading term of the dividend by the leading term of the divisor
2. Multiply the entire divisor by this quotient term
3. Subtract the result from the dividend
4. Bring down the next term
5. Repeat until no terms remain or the remainder has a lower degree than the divisor

Example:  Divide \( x^3 + 2x^2 - 5x - 6 \) by \( x + 2 \).

What is the quotient and remainder?

Solution:

Set up the long division:

Step 1: Divide \( x^3 \) by \( x \) to get \( x^2 \). Write \( x^2 \) above the division symbol.
Multiply \( x^2 \) by \( (x + 2) \): \( x^3 + 2x^2 \)
Subtract from the dividend: \( (x^3 + 2x^2) - (x^3 + 2x^2) = 0 \)

Step 2: Bring down \( -5x \). Now work with \( 0 - 5x = -5x \).
Divide \( -5x \) by \( x \) to get \( -5 \). Write \( -5 \) in the quotient.
Multiply \( -5 \) by \( (x + 2) \): \( -5x - 10 \)
Subtract: \( (-5x) - (-5x - 10) = 10 \)

Step 3: Bring down \( -6 \). Now work with \( 10 - 6 = 4 \).
Since 4 has degree 0 and \( x + 2 \) has degree 1, we cannot divide further.
The remainder is 4.

The quotient is \( x^2 - 5 \) and the remainder is 4, so the answer is \( x^2 - 5 + \frac{4}{x+2} \).

Example:  Divide \( 2x^3 - x^2 + 3x - 7 \) by \( x - 2 \).

What is the quotient?

Solution:

Divide \( 2x^3 \) by \( x \) to get \( 2x^2 \).
Multiply: \( 2x^2(x - 2) = 2x^3 - 4x^2 \)
Subtract: \( (2x^3 - x^2) - (2x^3 - 4x^2) = 3x^2 \)

Bring down \( 3x \). Now work with \( 3x^2 + 3x \).
Divide \( 3x^2 \) by \( x \) to get \( 3x \).
Multiply: \( 3x(x - 2) = 3x^2 - 6x \)
Subtract: \( (3x^2 + 3x) - (3x^2 - 6x) = 9x \)

Bring down \( -7 \). Now work with \( 9x - 7 \).
Divide \( 9x \) by \( x \) to get \( 9 \).
Multiply: \( 9(x - 2) = 9x - 18 \)
Subtract: \( (9x - 7) - (9x - 18) = 11 \)

The quotient is \( 2x^2 + 3x + 9 \) with remainder 11.

Applications of Polynomial Arithmetic

Polynomial arithmetic appears throughout algebra and in many real-world contexts. Understanding how to manipulate polynomials allows you to model and solve problems in science, engineering, economics, and computer science.

In physics, polynomial expressions describe motion under constant acceleration. For example, the position of an object might be given by \( s(t) = -16t^2 + 64t + 5 \), where adding or subtracting such expressions can represent combined motions.

In business, profit functions are often polynomial expressions derived by subtracting cost polynomials from revenue polynomials.

In geometry, the area and volume of composite shapes can be expressed as products of polynomial expressions representing dimensions.

Mastering polynomial arithmetic gives you the tools to simplify complex expressions, solve higher-degree equations, and understand the behavior of polynomial functions-all essential skills for advanced mathematics and its applications.