Introduction to Derivatives: Background You’ll Need 1

  • Add and subtract polynomials

Adding and Subtracting Polynomials

The process of adding and subtracting polynomials is a fundamental skill in algebra that involves combining like terms. Like terms are terms that have exactly the same variable factors raised to the same powers, or exponents.

When working with polynomials, it’s essential to identify like terms because these are the only terms that can be combined through addition or subtraction.

[latex]5{x}^{2}[/latex] and [latex]-2{x}^{2}[/latex] are like terms because they both contain the variable [latex]x[/latex] raised to the second power. Thus, combining them is straightforward: [latex]3{x}^{2}[/latex].

In contrast [latex]3x[/latex] and [latex]3{x}^{2}[/latex] are not like terms and therefore cannot be added. are not like terms since their exponents differ ([latex]x^1[/latex] vs. [latex]x^2[/latex]). They cannot be combined through addition or subtraction.

How To: Combining Polynomials using Addition of Subtraction

  1. Identify and Group Like Terms: Begin by scanning the polynomial expression for terms that have the same variables raised to the same powers. Group these terms together for simplification.
  2. Combine Like Terms: Add or subtract the coefficients (numerical parts) of like terms. The variable part will remain unchanged.
  3. Simplify and Write in Standard Form: Once the like terms are combined, ensure that the polynomial is written in standard form. 

The standard form of a polynomial starts with the term that has the highest power (degree) and proceeds in descending order of degree.

Add the following polynomials:

[latex]\left(12{x}^{2}+9x - 21\right)+\left(4{x}^{3}+8{x}^{2}-5x+20\right)[/latex]


Be careful When subtracting polynomials

When subtracting a polynomial from another, be careful to subtract each term in the second from the first. That is, use the distributive property to distribute the minus sign through the second polynomial.

[latex]\begin{array}{cc}\left(3x^2-2x+9\right)-\left(x^2-4x+5\right)\text{}\hfill &\text{Distribute the negative in front of the parenthesis} \hfill \\
3x^2-2x+9 -x^2 -\left(-4x\right) - 5\hfill & \text{Be careful when subtracting a negative}.\hfill \\ 3x^2 - x^2 -2x+4x+9-5\hfill & \text{Rearrange terms in descending order of degree} \hfill \\ 2x^2 +2x +4 \hfill & \text{Combine like terms}. \hfill \end{array}[/latex]

Subtract the following polynomials:

[latex]\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)[/latex]


Watch this video to see more examples of adding and subtracting polynomials.