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:

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

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\( −4x + 3x^2 −7 + 9x- 12x^2 −5x^4 \)

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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:

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

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\((2x+1)(3x^2−x+4) \)

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