Polynomial operations can be simplified by combining like terms: terms with the same variable and same exponents. For example, \( a^2b \) and \( -4a^2b\) would be alike and can be combined. To combine the terms, simply add the coefficients:
\( a^2b -4a^2b = -3a^2b \)
Below are some more examples of terms that are alike and some unalike
| Term | Like Term | Unlike Term |
|---|---|---|
| \(a\) | \(3a, -2a, \cfrac{1}{3}a\) | \(a^2, \cfrac{1}{a}, \sqrt{a}\) |
| \(a^2\) | \(-5a^2,\cfrac{1}{4}a^2, 0.6a^2\) | \(\cfrac{1}{a^2}, \sqrt{a^2}, a^3\) |
| \(ab\) | \(7ab,0.2ab,\cfrac{2}{3}ab,-ab\) | \(a^2b,\cfrac{1}{ab}, \sqrt{ab}\) |
| \(ab^2\) | \(4ab^2, \cfrac{ab^2}{7},0.4ab^2,-ab^2\) | \(a^2b,ab,\sqrt{ab^2},\cfrac{1}{ab^2}\) |
We can simplify \( (6k - 4) + (2k + 4) \) using the following steps.
First, remove the brackets and identify the like terms:
\( \textcolor{blue}{6k} - 4 + \textcolor{blue}{2k} + 4 \)
Next, reorder the terms collecting like terms:
\( \textcolor{blue}{6k} + \textcolor{blue}{2k} + 4 - 4 \)
Finally, collect the like terms:
\( \textcolor{blue}{8k} \)
\( (x^2 + 2x + 1) + (2x^2 + 4) \)
\( −4x + 3x^2 −7 + 9x- 12x^2 −5x^4 \)
The distributive property is used to multiply polynomials. First let's start with just one number to distribute:
\( 4 (x^2 - 3y) \)
Here, the \(4\) should be multiplied to every term within the brackets:
\( (4)(x^2) + (4)(- 3y) \)
Notice we used an addition sign between the new terms. Remember your integer rules for multipliation (same sign = positive, opposite sign = negative).
\( 4x^2 - 12y \)
To multiply polynomials, we need to multipy combinations of all the terms:
\( (3-x^3)(4y+3) \)
| \( (\colorbox{yellow}{3}-x^3)(\colorbox{yellow}{4y}+3) \) | \( (\colorbox{yellow}{3}-x^3)(4y+\colorbox{yellow}{3}) \) | \( (3\colorbox{yellow}{-x^3})(\colorbox{yellow}{4y}+3) \) | \( (3\colorbox{yellow}{-x^3})(4y+\colorbox{yellow}{3}) \) |
| \( 12y \) | \( 9 \) | \( -4x^3y \) | \( -3x^3 \) |
Then, add up the terms and collect like terms (if possible):
\( 12y + 9 -4x^3y -3x^3\)
Another way to approach the problem is to split up the first polynomial and multiply each term by the second polynomial:
\( (3-x^3)(4y+3) \)
\( (3)(4y+3) + (-x^3)(4y+3) \)
Now distribute just one number:
\( (12y)+(9) + (-4x^3y)+(-3x^3) \)
\( 12y + 9 -4x^3y -3x^3 \)
We get the same answer as before.
\((2x-18)(3x+3) \)
\((2x+1)(3x^2−x+4) \)