Expand and simplify
Algebraic expressions can be expanded - multiplied by one or more terms.
They may also be simplified – made shorter and simpler by collecting like terms.
Rules for multiplying in alegbra
- The X sign for multiplication is not needed.
\(6 \times k\) is written as \(6k\) (Remember that \(k\) means \(1k\))
- To multiply a term by a number, multiply the numbers
\(2 \times 5a = 10a\)
\(3 \times 2pq = 6pq\)
Remember the rules for multiplying positive and negative numbers:
- When two numbers with the same signs are multiplied the answer is positive
\(6p \times +7 = +42p\)
\(-5 \times -9m = +45m\)
- When two numbers with different signs are multiplied the answer is negative
\(3 \times -7q = -21q\)
\(-4 \times 2t = -8t\)
Example
Simplify \(5 \times 3q\)
Solution
Multiply the numbers and leave out the multiplication sign
Answer
\(15q\)
The expression \( 5 \times 3q\) has been simplified to give \(15q\)
Test yourself
Simplify \(2 \times c \times d\)
\(2 \times c \times d = 2cd\)
Example
Simplify \(3kj \times 4\)
Answer
\(12kj\)
An answer of \(12jk\) is also correct. The order of the letters is not important, but the number must come first.
Example
Simplify \(-3t \times -4t\)
Solution
Multiply \(-3\) by \(-4\) using the rule for numbers with the same sign\(-3 \times 4 = +12\)\(-3t \times -4t = +12 \times t \times t\)\(= 12t^2\)
Answer
\(12t^2\)
Expanding brackets
When more than one term is to be multiplied by a number, it is usual to put a bracket around these terms.
For example \(4(3m + n)\) is a shorter way to write \(4 \times 3m + 4 \times n\)Therefore \(4(3m + n) = 12m + 4n\)
The expression \( 4(3m + n)\) has been expanded to give \(12m + 4n\)
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Don't forget to multiply both terms in the bracket. A common mistake is to multiply the first term and forget about the second one.
Example
Expand \(3(7p - 2p)\)
Solution
The \(3\) outside the bracket is positive. There is no need to have a \(+\) in front of it.\(3(7p - 2q) = 3 \times 7p = 3 \times -2q\)\(= 21p - 6q\)
Answer
\(3(7p - 2q) = 21p - 6q\)
Test yourself
Expand \(5(6t - 7v + w)\)
Multiply each of the three terms by 5, using the rules to ensure that the signs are right!
Answer \(5(6t - 7v + w) = 30t - 35v + 5w\)
The expression \(30t - 35v + 5w\) cannot be simplified any further as there are not like terms to collect.
Test yourself
Expand \(-3(2f - 5g + 4)\)
Multiply each of the terms by \(-3\) using the rules to ensure that the signs are right!
Answer \(-3(2f - 5g + 4) = -6f + 15g - 12\)
Like terms
An expression is made up of a number of terms. It may be possible to simplify an expression by collecting like terms together. Like terms are easy to spot - they have the same letter or letters.
\(y\) Â Â Â \(3y\) Â Â Â \(-24y\) Â Â Â \(487y\) Â Â Â are all like terms.
\(3y\) and \(3xy\)   are not like terms but    \(3xy\) and \(5yx\)    are.
\(4y\) and \(9y^2\)    are not like terms but    \(4y^2\) and \(9y^2\)   are.
\(-7\)Â Â Â \(13\)Â Â Â \(127\) Â Â Â \(-46\)Â Â Â \(-10000\) Â Â Â are like terms.
All numbers are like terms.
Simplifying by collecting like terms
Expressions can be simplified by adding / subtracting like terms as required.
Rules for adding and subtracting terms:
- Only like terms can be added / subtracted
- Remember the rules for combining positive and negative numbers and terms
- When numbers have the same sign, keep that sign and add the numbers.
\(7 + 8 = +15\) (or just \(15\)) and \(7t + 8t = +15t\) (or just \(15t\))
\(-7 - 8 = -15\) and \(-7m - 8m = -15m\)
\(-7 - 8 -3 = -18\) and \(-7q - 8q - 3q = -18q\)
- When two numbers have different signs, use the sign in front of the bigger number and subtract the numbers.
\(12 - 10 = 2\) and \(12p - 10p = 2p (or +2p)\)
\(-11 + 8 = -3\) and \(-11k + 8k = -3k\)
\(8 - 9 = -1\) and \(8j - 9j = -j\)
\(-9 + 11 = +2\) and \(-9y + 11y = +2y\)
Example
Simplify \( 5t - 9t\)
Solution
The signs are different (\(5t\) means \(+5t\))
Use the sign in front of the bigger number and subtract the numbers.
\(9\) is bigger than \(5\) so use a \(-\) sign and subtract the numbers
Answer
\(5t - 9t = -4t\)
Example
Simplify \(-5y + 7 - 14y\)
Solution
\(-5y\) and \(-14y\) are like terms so we can combine these. Both terms have a \(-\) sign.
When numbers have the same sign, keep that sign and add the numbers.
\(-5y - 14y = -19y\)
That leaves the term \(+7\) which can be placed before or after the \(-19y\)
Answer
\(-5y + 7 - 14y = -19y + 7\)
or \(-5y + 7 - 14y = 7 - 19y\)
Both answers are equally correct
Test yourself
Simplify the expression \(6ac – 7c + ac - 3c\) by collecting like terms.
There are two types of like terms, terms in \(ac\) and terms in \(c\). Deal with them separately.
\(6ac + ac = 7ac\)
\(-7c - 3c = -10c\)
Answer \(6ac - 7c + ac - 3c = 7a - 10c\)
or
\(6ac - 7c + ac - 3c = -10c + 7ac\)
Both answers are equally correct.
Test yourself
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