Polynomial operations are based on two main ideas: combine like terms for addition and subtraction,
and use the distributive property for multiplication.
1. Like terms
Like terms have the same variable raised to the same power. For example:
\[
\begin{aligned}
3x^2 \text{ and } x^2
\end{aligned}
\]
are like terms, but \(x^2\) and \(x\) are not like terms.
2. Adding polynomials
To add polynomials, align terms with the same power and add their coefficients.
\[
\begin{aligned}
(3x^2+2x-5)+(x^2-7x+8)
&=
(3x^2+x^2)+(2x-7x)+(-5+8)\\
&=
4x^2-5x+3.
\end{aligned}
\]
3. Subtracting polynomials
To subtract polynomials, distribute the minus sign through the second polynomial,
then combine like terms.
\[
\begin{aligned}
(5x^3-2x+1)-(2x^3+4x^2-7)
&=
5x^3-2x+1-2x^3-4x^2+7\\
&=
3x^3-4x^2-2x+8.
\end{aligned}
\]
A common mistake is to change only the first sign after the subtraction sign.
The minus applies to every term in the second polynomial.
4. Multiplying polynomials
To multiply polynomials, every term in the first factor must multiply every term in the second factor.
\[
\begin{aligned}
(a+b)(c+d)
&=
ac+ad+bc+bd.
\end{aligned}
\]
This is the distributive property. For two binomials, it is often called FOIL.
5. Worked sample
Evaluate:
\[
\begin{aligned}
(3x^2+2x-5)+(x^2-7x+8)(2x+3).
\end{aligned}
\]
First multiply:
\[
\begin{aligned}
(x^2-7x+8)(2x+3)
&=
x^2(2x+3)-7x(2x+3)+8(2x+3)\\
&=
2x^3+3x^2-14x^2-21x+16x+24.
\end{aligned}
\]
Combine like terms in the product:
\[
\begin{aligned}
2x^3+3x^2-14x^2-21x+16x+24
&=
2x^3-11x^2-5x+24.
\end{aligned}
\]
Now add the remaining polynomial:
\[
\begin{aligned}
(3x^2+2x-5)+(2x^3-11x^2-5x+24)
&=
2x^3+(3x^2-11x^2)+(2x-5x)+(-5+24)\\
&=
2x^3-8x^2-3x+19.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{
(3x^2+2x-5)+(x^2-7x+8)(2x+3)
=
2x^3-8x^2-3x+19
}.
\end{aligned}
\]
6. Degree tracking
The degree of a polynomial is the largest exponent after the expression is simplified.
For example:
\[
\begin{aligned}
2x^3-8x^2-3x+19
\end{aligned}
\]
has degree \(3\), because the largest exponent is \(3\).
7. Multiplication and degree
When multiplying nonzero polynomials, degrees usually add:
\[
\begin{aligned}
\deg(PQ)
&=
\deg(P)+\deg(Q).
\end{aligned}
\]
For example:
\[
\begin{aligned}
\deg(x^2-7x+8)&=2,\\
\deg(2x+3)&=1,\\
\deg\big((x^2-7x+8)(2x+3)\big)&=3.
\end{aligned}
\]
8. Vertical alignment
Vertical alignment helps students see which terms combine.
For addition, align the powers:
\[
\begin{aligned}
&(3x^2+2x-5)\\
+\;&(x^2-7x+8)\\
=\;&4x^2-5x+3.
\end{aligned}
\]
Each column contains like terms.
9. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
10. Common mistakes
- Combining unlike terms such as \(x^2\) and \(x\).
- Forgetting to change every sign when subtracting a polynomial.
- Multiplying only the first terms and forgetting the remaining products.
- Dropping negative signs during multiplication.
- Forgetting to combine like terms after expansion.
- Reporting the degree before simplifying the final polynomial.
Key idea: add and subtract by aligning like powers; multiply by distributing every term, then combine like terms.
