# Operations with Polynomials

## Collecting Like Terms

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:

$$\textcolor{blue}{8k}$$

Simplify the following:

$$(x^2 + 2x + 1) + (2x^2 + 4)$$

$$−4x + 3x^2 −7 + 9x- 12x^2 −5x^4$$

## Distributive Property

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.

Simplify the following:

$$(2x-18)(3x+3)$$

$$(2x+1)(3x^2−x+4)$$