Lesson

We've already learned how to simplify expressions with grouping symbols. To distribute an expression like $3\left(x+2\right)$3(`x`+2) or $5\left(2y-1\right)$5(2`y`−1) we use the distributive property:

The distributive property

To distribute an expression of the form $a\left(b+c\right)$`a`(`b`+`c`), we use the property:

$a\left(b+c\right)$a(b+c) |
$=$= | $a\times b+a\times c$a×b+a×c |

$=$= | $ab+ac$ab+ac |

So far we have used the distributive property to simplify expressions involving multiplication of constants with variables. Now we will look at how to use the distributive property to simplify expressions involving multiplication of variables. We will need to use the product of powers property.

**Distribute:** $5x\left(6x^6-3y\right)$5`x`(6`x`6−3`y`).

**Think: **We'll distribute the parentheses using the distributive property:

To evaluate the products $5x\times6x^6$5`x`×6`x`6, we can use the product of powers property:

The product of powers property

To multiply terms with like bases, (e.g. $a$`a` and $a$`a`) we use the rule:

$a^m\times a^n$am×an |
$=$= | $a^{m+n}$am+n |

For example,

$x\times x^2$x×x2 |
$=$= | $x^{1+2}$x1+2 |

$=$= | $x^3$x3 |

**Do:**

$5x\left(6x^6-3y\right)$5x(6x6−3y) |
$=$= | $\left(5x\right)\left(6x^6\right)-\left(5x\right)\left(3y\right)$(5x)(6x6)−(5x)(3y) |

$=$= | $30x^7-15xy$30x7−15xy |

Distribute the following:

$r\left(r+5\right)$`r`(`r`+5)

Distribute $6u^7\left(9u^7+9u^6\right)$6`u`7(9`u`7+9`u`6)

We can take the distributive property one step further in two ways. First, we may be required to simplify the expression after distributing. Secondly, we can extend beyond a product of a monomial and binomial to a monomial and a polynomial.

Let's look at simplification which is possible only after distributing.

The order of operations

The order of operations when you are simplifying algebraic expressions is:

**Step 1**: Do operations inside grouping symbols such as round parentheses (...), square parentheses [...] and braces {...}.

**Step 2**: Distribute sets of parentheses using the distributive property.

**Step 3**: Do multiplication and division going from left to right.

**Step 4**: Do addition and subtraction going from left to right.

Let's look at how to simplify an expression with more than one set of parentheses.

Distribute and simplify: $3r\left(-5r+4s\right)-6r\left(4r-2s\right)$3`r`(−5`r`+4`s`)−6`r`(4`r`−2`s`)

**Think**: We will need to use the distributive property twice. Once shown in green below and the other shown in red below.

**Do:** When we distribute this, we get:

$-15r^2+12rs-24r^2+12rs$−15`r`2+12`r``s`−24`r`2+12`r``s`

Then we can collect the like terms and write our simplified answer:

$24rs-39r^2$24`r``s`−39`r`2

**Reflect**: These are not like terms, so this can't be simplified any further.

**Simplify: **$3u\left(5u-9v\right)-12uv$3`u`(5`u`−9`v`)−12`u``v`

**Think**: The order of operations requires us to look at the parentheses first, then do any multiplication before addition/subtraction.

**Do**:

Step 1: Distribute the multiplication of $3u$3`u` across the parentheses using the distributive property:

$3u\left(5u-9v\right)-12uv=\left(3u\right)\left(5u\right)+\left(3u\right)\left(-9v\right)-12uv$3`u`(5`u`−9`v`)−12`u``v`=(3`u`)(5`u`)+(3`u`)(−9`v`)−12`u``v`

Step 2: Evaluate the products. We will use the product of powers property $a^m\cdot a^n=a^{m+n}$`a``m`·`a``n`=`a``m`+`n`.

$\left(3u\right)\left(5u\right)+\left(3u\right)\left(-9v\right)-12uv=15u^2-27uv-12uv$(3`u`)(5`u`)+(3`u`)(−9`v`)−12`u``v`=15`u`2−27`u``v`−12`u``v`

Step 3: Combine like terms. There are two terms in $uv$`u``v` so we can combine these by adding the coefficients.

$15u^2-27uv-12uv=15u^2-39uv$15`u`2−27`u``v`−12`u``v`=15`u`2−39`u``v`

So our simplified answer is $15u^2-39uv$15`u`2−39`u``v`.

**Reflect**: We need to be sure that we are only distributing the monomial to the terms in the parentheses and not the terms outside of the grouping symbol.

Distribute the following and simplify: $8u\left(10u+v\right)-5$8`u`(10`u`+`v`)−5

Distribute and simplify $2a\left(5a^2+2a+3\right)$2`a`(5`a`2+2`a`+3)

Distribute and simplify the following expression: $9x\left(4x+8y\right)-2x\left(4y-4x\right)$9`x`(4`x`+8`y`)−2`x`(4`y`−4`x`)

Multiply polynomials of degree one and degree two

Rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property