3(n + 4) means three lots of (n + 4). That is 3n and 3 lots of 4.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Every term, not just the first
3(n + 4) is 3n + 12. Writing 3n + 4 leaves two of the three lots of 4 behind.
Why the area model shows it
A rectangle 3 tall and (n + 4) wide splits into a 3-by-n piece and a 3-by-4 piece.
The opposite of Chapter 3
Pulling a common factor out was this run backwards. Expanding puts it back in.
Multiply everything inside
3(n + 4) = 3n + 12. The 3 multiplies both terms inside the brackets, not just the first. This is the distributive property, met in Grade 3 as a rectangle and now applied to letters.
The rectangle is still the picture
Draw a rectangle 3 tall and (n + 4) wide. Split it into a 3-by-n part and a 3-by-4 part: areas 3n and 12. The same picture that explained 3 × 47 explains 3(n + 4). Nothing new has happened except that one length is a letter.
When the multiplier is negative
−2(n − 3) = −2n + 6. Every term inside gets multiplied, sign and all, and multiplying two negatives gives a positive. This is where most expansion errors live, so slow down and handle each term separately.
Substitute to check
With n = 5, 3(n + 4) = 27 and 3n + 12 = 27. Agreement at a test value is strong evidence the expansion was done correctly, and it takes seconds. Use a value other than 0 or 1, which can hide errors.
Step 2: Try It Yourself
Tap and try it out.
9 + 9 + 9 = 27
3 × 9 = 27
3 × 9 = (3 × 4) + (3 × 5) = 12 + 15 = 27
The array is cut into 3 × 4 and 3 × 5. Two easier facts that add to the same total.
Step 3: Watch an Example
One step at a time.
Watch Leo Expand 4(n + 3)
Leo multiplies a bracket out.
- Step 1
He multiplies the 4 by the first term: 4 × n = 4n.
Step 4: Your Turn
Practice makes it stick.
The Check
Problem 1 of 2
Expand 5(n + 2), then evaluate when n is 3.
The Constant
Problem 2 of 2
In 6(n + 5), what number does the 6 multiply the 5 by to give?
Multiply It Out
1 of 4
Expand 3(n + 5). What is the number term?
2 of 4
Expand 7(n + 2). What is the number term?
3 of 4
Is 3(n + 4) the same as 3n + 4?
4 of 4
Evaluate 2(n + 6) when n is 4.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Expand 5(n + 3). What is the number term?
What You Learned
- A number outside a bracket multiplies every term inside it.
- The area model shows why: the rectangle splits into two pieces.
- Expanding and pulling out a common factor are the same move in opposite directions.