Lesson 53: Adding and Subtracting Polynomials

New Concept

A polynomial is a monomial or the sum or difference of monomials.

What’s next

Next, you’ll learn the basic operations for these expressions: how to add and subtract polynomials by combining like terms.

Property

The degree of a monomial is the sum of the exponents of the variables in the monomial. A constant has a degree of 0.

Examples

• The degree of $8a^3b^6c$ is $3 + 6 + 1 = 10$. Remember that $c$ is the same as $c^1$.
• The degree of $15x^4y^5z^2$ is $4 + 5 + 2 = 11$. Just sum the variable exponents!
• The degree of $50^2x^5y^3$ is $5 + 3 = 8$. The exponent on the number $50$ is a trap; ignore it!

Explanation

Think of this as finding the 'power level' of a term! To find the degree, you simply add up the exponents of all the variables. Be careful to ignore any exponents on the coefficients (the numbers in front). The game is all about counting the total power of the variables, nothing else matters for the degree!

Property

The standard form of a polynomial is a form of a polynomial where terms are ordered from greatest to least degree.

Examples

• To write $5y^3 + y^6$ in standard form, you reorder it to $y^6 + 5y^3$ since degree 6 is greater than degree 3.
• Given $7ab^4 - 12 + 3a^5b^2c$, the standard form is $3a^5b^2c + 7ab^4 - 12$. The degrees are 8, 5, and 0.
• The polynomial $-5x^4y^2 + 12x^3y - 8xy$ is already in standard form, with degrees 6, 4, and 2 in descending order.

Explanation

Putting a polynomial in standard form is like creating a power-ranking of its terms. First, find the degree of each individual term. Then, simply arrange them in a line from the highest degree down to the lowest. The coefficient of that first, most powerful term gets the special title of 'leading coefficient.'

Property

Like terms, such as $-8x^4$ and $x^4$, have the same variables raised to the same powers.

Examples

• The terms $7a^2b$ and $-2a^2b$ are like terms because their variable part, $a^2b$, is identical.
• The terms $4x^3y^2$ and $4x^2y^3$ are NOT like terms. The exponents on $x$ and $y$ are different.
• To simplify $(4x^2 + 6x) + (3x^2 - 2x)$, you combine the like terms $4x^2$ and $3x^2$, and then $6x$ and $-2x$.

Explanation

Imagine like terms are identical twins in the algebra world. They must have the exact same variables with the exact same exponents—their 'variable DNA' has to be a perfect match! The numbers in front (coefficients) can be different, but that's it. Only these identical twins can be combined when you add or subtract polynomials.