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Math Lesson 6.2.2 - How to expand brackets when a number multiplies the Expression inside them
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Welcome to our Math lesson on How to expand brackets when a number multiplies the Expression inside them, this is the second lesson of our suite of math lessons covering the topic of Expanding Brackets, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Expanding Brackets When a Number Multiplies the Expression inside Them
This rule is for expressions of the form
and
Such brackets are expanded by applying the property where:
A number that multiplies an expression inside brackets multiplies each term contained in the brackets separately when the brackets are removed. In this case, we say the expression is expanded according to the following scheme.
Obviously, when the number that multiplies the expression contained in the brackets is positive, all terms preserve their sign; otherwise (when the number that multiplies the expression in the brackets is negative) all terms change sign.
It is obvious that expanding brackets is carried out based on the distributive property of multiplication
we discussed in the first chapter of this math course.
When completing operations with algebraic terms, we must consider the sign rules of multiplication (and division), these are displayed in the table below.
| Sign of a | Sign of b | Sign of a × b (a ÷ b if b ≠ 0) |
|---|---|---|
| + | + | + |
| + | - | - |
| - | + | - |
| - | - | + |
Example 2
Calculate the value of the following algebraic expression
for x = 5 and y = 4 in two ways (one by direct substitution and the other by expanding the brackets).
Solution 2
First Method (by direct substitution):
= -5 ∙ (3 ∙ 5 - 2 ∙ 4) + 3 ∙ (5 + 4) - 1
= -5 ∙ (15 - 8) + 3 ∙ (5 + 4) - 1
= -5 ∙ 7 + 3 ∙ 9 - 1
= -35 + 27 - 1
= -8 - 1
= -9
Second Method (by expanding brackets first):
= -5 ∙ 3x - 5 ∙ (-2y) + 3 ∙ x + 3 ∙ y - 1
= -15x + 10y + 3x + 3y - 1
= -12x + 13y - 1
Now, substituting x = 5 and y = 4 yields
= -60 + 52 - 1
= -8 - 1
= -9
which is the same result we found through the direct substitution method.
Substituting the values assigned to variables is not always possible, as often there are no values assigned to them. For this reason, the second method based on expanding brackets is more appropriate (and inclusive), as it provides us with the simplest version of an algebraic expression, the value of which can be easily calculated for certain values of its variables if necessary.
Remarks!
- The number that multiplies the expression inside brackets may come before or after them; this does not alter the value of the expression. This is because multiplication has the commutative property, i.e. we can change the position of factors without affecting the product.
- We can multiply an expression inside a brackets by a letter (variable) as well. If the variable is unknown, it is assumed as positive in the sense that we leave the intermediate signs as they are. Look at this example.
Example 3
- Simplify the following algebraic expression: 2x ∙ (3 - 5y) + 4 ∙ (xy - 1) + 3xy
- Calculate the value of this expression for x = 1 and y = -2
Solution 3
- We have 2x ∙ (3 - 5y) + 4 ∙ (xy - 1) + 3xy
2x ∙ 3 - 2x ∙ 5y + 4 ∙ xy + 4 ∙ (-1) + 3xy
= 6x - 10xy + 4xy - 4 + 3xy
= 6x - 3xy - 4 - Substituting x = 1 and y = -2 in the last expression yields 6x - 3xy - 4
= 6 ∙ 1 - 3 ∙ 1 ∙ (-2) - 4
= 6 + 6 - 4
= 12 - 4
= 8
More Expanding Brackets Lessons and Learning Resources
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- Number Multiplies Feedback. Helps other - Leave a rating for this number multiplies (see below)
- Expressions Math tutorial: Expanding Brackets. Read the Expanding Brackets math tutorial and build your math knowledge of Expressions
- Expressions Video tutorial: Expanding Brackets. Watch or listen to the Expanding Brackets video tutorial, a useful way to help you revise when travelling to and from school/college
- Expressions Revision Notes: Expanding Brackets. Print the notes so you can revise the key points covered in the math tutorial for Expanding Brackets
- Expressions Practice Questions: Expanding Brackets. Test and improve your knowledge of Expanding Brackets with example questins and answers
- Check your calculations for Expressions questions with our excellent Expressions calculators which contain full equations and calculations clearly displayed line by line. See the Expressions Calculators by iCalculator™ below.
- Continuing learning expressions - read our next math tutorial: Special Algebraic Identities Obtained through Expanding
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