Adding and Subtracting Polynomials Lesson

Before we get into adding and subtracting polynomials, it is important to understand the definition of a "term". A term is a number, variable, or the product of a number and one or more variables raised to powers. Terms are separated by "+" or "-" signs. In our above picture, we have the algebraic expression:
7x² + 6x - 9
The 7x², 6x, and 9 are terms. Notice how they are separated by "+" or "-" signs.
"Like Terms" occur when two or more terms have exactly the same variable parts. This means the variable(s) is(are) the same and raised to the same power(s).
5x², 9x² » Like Terms: each has the variable x raised to the 2nd power.
9x³y², 15x³y² » Like Terms: each has the variable x raised to the 3rd power and the variable y raised to the 2nd power.
2x⁴, 13x » Not Like Terms: each has the variable x, but our exponents are not the same.
11x³, 14y³ » Not Like Terms: although the exponents are the same, the variables are different.
We can add or subtract "like terms" by keeping the variable part the same and performing operations with the coefficients. We are just using the distributive property.
3x - 5x = (3 - 5)x = -2x
13x² + 9x² = (13 + 9)x² = 22x²

What is a Polynomial

A polynomial is either a single term or a finite sum of terms where all variables have whole number exponents and there are no variables in any denominator. A polynomial is the simplest type of algebraic expression.
Examples of Polynomials: $$11$$ $$-2x^3-5x$$ $$12x^2-7x+1$$ Examples of Non-Polynomials: $$\frac{2x - 5}{x^2}$$ $$5x^{-4}+ 1$$ $$\sqrt{7 - x}$$ Some polynomials occur so often, that we give them special names. A monomial is a polynomial with only one term.
Examples of Monomials:
8x⁷
12x⁴
-6x⁵y³
A binomial is a polynomial with two terms.
Examples of Binomials:
9x⁴ - 2
19x³ - 4x²
-11x⁵ + 7
Examples of Trinomials:
25x³ + 2x - 14
-5x² - 12x + 1
3x⁹ + 4x⁵ + 6

Standard Form of a Polynomial

A polynomial is written in standard form when the powers are in descending order.
9x + 7x⁴ - 2 + 12x² - 3x³
We can rewrite this polynomial in standard form:
7x⁴ - 3x³ + 12x² + 9x - 2
Notice how the first or leftmost term "7x⁴" has the largest exponent on x. Then each position that follows has the next largest exponent on x. The final term (rightmost) is a 2. We can think about this as 2x⁰ since x⁰ is 1 and 2 • 1 is 2. Generally speaking, we are expected to write any polynomial answer in standard form.

The Degree of a Polynomial

The degree of a term in a polynomial is the sum of the exponents on the variable(s) of the term.
12x » The degree is 1
4x⁹ » The degree is 9
-10x⁵y³ » The degree is 8
7x³y⁷z¹⁴ » The degree is 24
4 » The degree is 0 (4x⁰ = 4 • 1 = 4)
The largest degree of any non-zero term in the polynomial is the degree of the polynomial.
7x² - x + 1 » degree of 2
x³y² + 4xy - 3 » degree of 5
x¹⁸y³z⁵ - 9x²y³z + 14 » degree of 26

Adding Polynomials

To add polynomials, we simply combine like terms. Let's take a look at a few examples.
Example 1: Simplify each. $$(8x^2+9x-9) + (13x^2+4x- 1)$$ Since we have addition, we can simply drop the parentheses. Let's rewrite the problem and arrange our like terms next to each other: $$8x^2 + 13x^2 + 9x + 4x - 9 - 1$$ Now, we can combine like terms: $$21x^2 + 13x - 10$$ Example 2: Simplify each. $$(12x^4 - 6x^2 + 12x) + (-11x^2 + 11x + 7) + (-4x^4 + 10x^2 - 12x)$$ Since we have addition, we can simply drop the parentheses. Let's rewrite the problem and arrange our like terms next to each other: $$12x^4 - 4x^4 - 6x^2 - 11x^2 + 10x^2 + 12x + 11x - 12x + 7$$ Now, we can combine like terms: $$8x^4 - 7x^2 + 11x + 7$$ Example 3: Simplify each. $$(4x^4y^2 + 2x^2y + 5x^3 + 2x) + (-6x^3 + 5y^3 + 8x^2 + 7x) + (x^2 - 2x^2y)$$ Since we have addition, we can simply drop the parentheses. Let's rewrite the problem and arrange our like terms next to each other: $$4x^4y^2 + 2x^2y - 2x^2y + 5x^3 - 6x^3 + 5y^3 + 8x^2 + x^2 + 2x + 7x$$ Now, we can combine like terms: $$4x^4y^2 -x^3 + 5y^3 + 9x^2 + 9x$$

Subtracting Polynomials

When we subtract one polynomial from another, we change the subtraction sign to addition and change the sign of each term of the polynomial that is being subtracted away. In some cases, we will see tutorials where the sign is changed from "-" to "+" and then a "-1" is written outside of the polynomial that is being subtracted away. This "-1" just serves as a reminder. When we distribute the "-1" to each term inside of the parentheses, we are changing the sign of each term.
Example 4: Simplify each. $$(2x^4 + 2x^2 - 4) - (x^4 + 7x^2 - 6) - (-3x^4 + 2x^2)$$ Let's begin by changing each subtraction into addition. We will also change each term that is being subtracted away into its opposite. To make this process clear, let's change the sign from "-" to "+" and place a "-1" outside of the parentheses: $$(2x^4 + 2x^2 - 4) + (-1)(x^4 + 7x^2 - 6) + (-1)(-3x^4 + 2x^2)$$ Now, let's distribute the -1 to each term inside of the parentheses. This will change the sign of each term that is being subtracted away: $$2x^4 + 2x^2 - 4 - x^4 - 7x^2 + 6 + 3x^4 - 2x^2$$ Now, we just have addition. We can rewrite the problem and arrange our like terms next to each other: $$2x^4 - x^4 + 3x^4 + 2x^2 - 7x^2 - 2x^2 - 4 + 6$$ Now, we can combine like terms: $$4x^4 - 7x^2 + 2$$ Example 5: Simplify each. $$(2x^3y^2 + 4 - 6y^4) - (3x^3y^2 - 3x^4y^4 - 4) - (2 - 7y^4 + 5x^3y^2)$$ Let's begin by changing each subtraction into addition. We will also change each term that is being subtracted away into its opposite. To make this process clear, let's change the sign from "-" to "+" and place a "-1" outside of the parentheses: $$(2x^3y^2 + 4 - 6y^4) + (-1)(3x^3y^2 - 3x^4y^4 - 4) + (-1)(2 - 7y^4 + 5x^3y^2)$$ Now, let's distribute the -1 to each term inside of the parentheses. This will change the sign of each term that is being subtracted away: $$2x^3y^2 + 4 - 6y^4 - 3x^3y^2 + 3x^4y^4 + 4 - 2 + 7y^4 - 5x^3y^2$$ Now, we just have addition. We can rewrite the problem and arrange our like terms next to each other: $$3x^4y^4 + 2x^3y^2 - 3x^3y^2 - 5x^3y^2 - 6y^4 + 7y^4 + 4 + 4 - 2$$ Now, we can combine like terms: $$3x^4y^4 - 6x^3y^2 + y^4 + 6$$

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