Polynomials can can be simplified by combining like terms, or terms with the same variables and exponents.
For example, \( a^2b \) and \( -4a^2b\) would be considered like terms and can be combined. To collect the terms, simply add or subtract the coefficients:
Below are some more examples of terms that are like and unlike:
| Term | Like Term | Unlike Term |
|---|---|---|
| \(a\) | \(3a, -2a, \cfrac{1}{3}a\) | \(a^2, \cfrac{1}{a}, \sqrt{a}\) |
| \(a^2\) | \(-5a^2,\cfrac{1}{4}a^2, 0.6a^2\) | \(\cfrac{1}{a^2}, \sqrt{a^2}, a^3\) |
| \(ab\) | \(7ab,0.2ab,\cfrac{2}{3}ab,-ab\) | \(a^2b,\cfrac{1}{ab}, \sqrt{ab}\) |
| \(ab^2\) | \(4ab^2, \cfrac{ab^2}{7},0.4ab^2,-ab^2\) | \(a^2b,ab,\sqrt{ab^2},\cfrac{1}{ab^2}\) |
Simplify the expression \((6k - 4) + (2k + 4)\).
We can simplify \( (6k - 4) + (2k + 4) \) using the following steps.
First, we can remove the brackets and identify the like terms:
Next, we can identify the like terms and reorder them:
Finally, we can collect the like terms:
Therefore, we can determine that the expression simplified is \(\boldsymbol{8k}\).
\( (x^2 + 2x + 1) + (2x^2 + 4) \)
First, we can remove the brackets and identify the like terms:
\(\textcolor{blue}{x}^\textcolor{red}{2} + \textcolor{green}{2x} + 1 + \textcolor{blue}{2x}^\textcolor{red}{2} + 4\)
Next, we can identify the like terms and reorder them:
\(\textcolor{blue}{x}^\textcolor{red}{2} +\textcolor{green}{2x}^\textcolor{red}{2} + \textcolor{blue}{2x} + 1 + 4\)
Finally, we can collect the like terms:
\(\textcolor{blue}{3x}^\textcolor{red}{2} + \textcolor{green}{2x} + 5\)
Therefore, we can determine that the expression simplified is \(\boldsymbol{3x^2 + 2x + 5}\).
\( −4x + 3x^2 −7 + 9x- 12x^2 −5x^4 \)
First, we can identify the like terms and reorder them:
\(−5x^{4}+\textcolor{blue}{3x}^\textcolor{red}{2}−\textcolor{blue}{12x}^\textcolor{red}{2}+\textcolor{green}{9x}-\textcolor{green}{4x}−7\)
Finally, we can collect the like terms:
\(−5x^{4} -\textcolor{blue}{9x}^\textcolor{red}{2}+ \textcolor{green}{5x} − 7\)
Therefore, we can determine that the expression simplified is \(\boldsymbol{−5x^{4} - 9x^2 + 5x − 7}\).