2.02 Multiplying a polynomial by a monomial
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(2y−1) we use 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.
Worked example
Question 1
Distribute: $5x\left(6x^6-3y\right)$5x(6x6−3y).
Think: We'll distribute the parentheses using the distributive property:
To evaluate the products $5x\times6x^6$5x×6x6, we can use 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 |
Practice questions
Question 2
Distribute the following:
$r\left(r+5\right)$r(r+5)
Question 3
Question 4
Distribute $6u^7\left(9u^7+9u^6\right)$6u7(9u7+9u6)
Distributing with polynomials and further simplification
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 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.
Worked example
Question 5
Distribute and simplify: $3r\left(-5r+4s\right)-6r\left(4r-2s\right)$3r(−5r+4s)−6r(4r−2s)
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:
$-15r^2+12rs-24r^2+12rs$−15r2+12rs−24r2+12rs
Then we can collect the like terms and write our simplified answer:
$24rs-39r^2$24rs−39r2
Reflect: These are not like terms, so this can't be simplified any further.
Question 6
Simplify: $3u\left(5u-9v\right)-12uv$3u(5u−9v)−12uv
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$3u 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$3u(5u−9v)−12uv=(3u)(5u)+(3u)(−9v)−12uv
Step 2: Evaluate the products. We will use the product of powers property $a^m\cdot a^n=a^{m+n}$am·an=am+n.
$\left(3u\right)\left(5u\right)+\left(3u\right)\left(-9v\right)-12uv=15u^2-27uv-12uv$(3u)(5u)+(3u)(−9v)−12uv=15u2−27uv−12uv
Step 3: Combine like terms. There are two terms in $uv$uv so we can combine these by adding the coefficients.
$15u^2-27uv-12uv=15u^2-39uv$15u2−27uv−12uv=15u2−39uv
So our simplified answer is $15u^2-39uv$15u2−39uv.
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.
Practice questions
Question 7
Distribute the following and simplify: $8u\left(10u+v\right)-5$8u(10u+v)−5
Question 8
Distribute and simplify $2a\left(5a^2+2a+3\right)$2a(5a2+2a+3)
question 9
Distribute and simplify the following expression: $9x\left(4x+8y\right)-2x\left(4y-4x\right)$9x(4x+8y)−2x(4y−4x)