Expanding a polynomial means rewriting a product or power as a sum of terms.
The main idea is the distributive property: every term in one factor must multiply every term in the other factor.
1. The distributive property
The basic rule is:
\[
\begin{aligned}
a(b+c)
&=
ab+ac.
\end{aligned}
\]
Example:
\[
\begin{aligned}
3(x+5)
&=
3x+15.
\end{aligned}
\]
2. Multiplying two binomials
For two binomials, each term in the first parentheses multiplies each term in the second parentheses:
\[
\begin{aligned}
(a+b)(c+d)
&=
ac+ad+bc+bd.
\end{aligned}
\]
This is often called FOIL:
- F: first terms
- O: outer terms
- I: inner terms
- L: last terms
3. Worked example: \((x+4)(2x^2-3x+5)\)
Distribute \(x\) across the trinomial:
\[
\begin{aligned}
x(2x^2-3x+5)
&=
2x^3-3x^2+5x.
\end{aligned}
\]
Distribute \(4\) across the trinomial:
\[
\begin{aligned}
4(2x^2-3x+5)
&=
8x^2-12x+20.
\end{aligned}
\]
Add the results:
\[
\begin{aligned}
(x+4)(2x^2-3x+5)
&=
2x^3-3x^2+5x+8x^2-12x+20.
\end{aligned}
\]
Combine like terms:
\[
\begin{aligned}
2x^3-3x^2+8x^2+5x-12x+20
&=
2x^3+5x^2-7x+20.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{
(x+4)(2x^2-3x+5)
=
2x^3+5x^2-7x+20
}.
\end{aligned}
\]
4. Combining like terms
Like terms have the same variables with the same powers.
For example, \(-3x^2\) and \(8x^2\) are like terms:
\[
\begin{aligned}
-3x^2+8x^2
&=
5x^2.
\end{aligned}
\]
Similarly:
\[
\begin{aligned}
5x-12x
&=
-7x.
\end{aligned}
\]
5. Polynomial powers
A power such as \((x+2)^3\) means repeated multiplication:
\[
\begin{aligned}
(x+2)^3
&=
(x+2)(x+2)(x+2).
\end{aligned}
\]
First:
\[
\begin{aligned}
(x+2)(x+2)
&=
x^2+4x+4.
\end{aligned}
\]
Then:
\[
\begin{aligned}
(x^2+4x+4)(x+2)
&=
x^3+6x^2+12x+8.
\end{aligned}
\]
6. Binomial theorem
For binomial powers, the binomial theorem gives:
\[
\begin{aligned}
(a+b)^n
&=
\sum_{k=0}^{n}
\binom{n}{k}
a^{n-k}b^k.
\end{aligned}
\]
For example:
\[
\begin{aligned}
(a+b)^4
&=
a^4+4a^3b+6a^2b^2+4ab^3+b^4.
\end{aligned}
\]
The coefficients \(1,4,6,4,1\) come from Pascal’s triangle.
7. Trinomial and multinomial expansion
A trinomial power such as \((x+y+1)^3\) can be expanded by repeated multiplication
or by the multinomial theorem. Repeated multiplication means:
\[
\begin{aligned}
(x+y+1)^3
&=
(x+y+1)(x+y+1)(x+y+1).
\end{aligned}
\]
Each term in each factor must multiply every matching term in the other factors.
8. Degree of an expanded polynomial
The degree of a term is the sum of the exponents in that term.
The degree of the polynomial is the largest degree among its terms.
Example:
\[
\begin{aligned}
2x^3+5x^2-7x+20
\end{aligned}
\]
has degree \(3\), because the largest power of \(x\) is \(3\).
9. Checking an expansion
To check an expansion, you can substitute a simple value for the variable into both the original expression
and the expanded expression. They should give the same result.
For example, using \(x=1\):
\[
\begin{aligned}
(1+4)(2(1)^2-3(1)+5)
&=
5(4)
=
20.
\end{aligned}
\]
Expanded form:
\[
\begin{aligned}
2(1)^3+5(1)^2-7(1)+20
&=
2+5-7+20\\
&=
20.
\end{aligned}
\]
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Multiplying only the first terms and forgetting the other distributed products.
- Forgetting negative signs, especially in terms like \(-3x\).
- Writing \((x+2)^2=x^2+4\), which incorrectly skips the middle term.
- Combining unlike terms such as \(x^2\) and \(x\).
- Dropping a term during a trinomial expansion.
- Forgetting that adjacent parentheses mean multiplication.
Key idea: multiply every required pair of terms, then combine only terms with exactly matching variable powers.
