Algebra : Expansion Within Brackets
This video course teaches how to expand expressions within brackets. Students will learn to multiply and simplify terms step by step, building confidence in handling more complex algebraic expressions.
Related Lessons:
Reduce Error When Multiplying Algebraic Terms
Many students make careless mistakes when multiplying algebraic terms, but these can be entirely avoided with the right approach. By multiplying the sign, coefficient, and algebraic part separately, students can ensure accurate results every time. For example, to evaluate \((-2xy)\times(3yz)\):
- Multiple the sign: \(-\) and \(+\) to get \(-\)
- Multiple the coefficient: 2 and 3 to get 6
- Multiple the algebra: \(xy\) and \(yz\) to get \(xy^2z\)
Combining the three items, we will get the answer \(-6xy^2z\).
Four Examples On Multiplication
Now, let us go through 4 examples of algebraic multiplication. Students are encouraged to try solving the questions on their own. For students who need step by step guidance, students can view the video solution at the end.
Example 1:
Expand \(7y\times\left(-6y\right)\)
Solution:
\[
\begin{align*}
7y\times\left(-6y\right)&=-\left(7\times6\right)\left(y\times{y}\right)\\
&=-42{y}^2
\end{align*}
\]
Example 2:
Expand \(\left(-5a\right)\times\left(-4b\right)\)
Solution:
\(\left(-5a\right)\times\left(-4b\right)\)\(=\left(-\times-\right)\left(5\times4\right)\left(a\times{b}\right)\)
\(=20ab\)
Example 3:
Expand \(\left(-3a\right)\times4ab\)
Solution:
\(
\left(-3a\right)\times\left(4ab\right)\)
\(=\left(-\times+\right)\left(3\times4\right)\left(a\times{ab}\right)\)
\(=-12{a}^2b\)
Example 4:
Expand \(3\times\left(-11a\right)\)
Solution:
\[
\begin{align*}
3\times\left(-11a\right)&=\left(+\times-\right)\left(3\times11\right)\left(a\right)\\
&=-33a
\end{align*}
\]
Simple Algebra Expansion Within Brackets
Next, we will learn how to expand algebraic expressions involving two brackets using the rainbow expansion method. Students will see how to multiply each term in the first bracket by each term in the second bracket step by step, in the sequence 1-1, 1-2, 2-1, 2-1 format. For students who needed step-by-step guidance, view the video presentation at the end.
Example 1:
Expand \(-\left(4x-3\right)\)
Solution:
\[
\begin{align*}
-\left(4x-3\right)&=\left(-\right)\left(4x\right)-\left(-3\right)\\
&=-4x+3
\end{align*}
\]
Example 2:
Expand \(-(2x^2-3x+4)\)
Solution:
\(
-\left(2x^2-3x+4\right)\)
\(=-1\left(2x^2-3x+4\right)\)
\(=-1\left(2x^2\right)-1\left(-3x\right)-1\left(+4\right)\)
\(=-2x^2+3x-4\)
Example 3:
Expand \(-2(5x-1)\)
Solution:
\[
\begin{align*}
-2(5x-1)&=\left(-2\right)\left(5x\right)-2\left(-1\right)\\
&=-10x+2
\end{align*}
\]
Rainbow Expansion For Two Or Three Brackets
Next, we will tackle something more advanced: expanding algebraic expressions with more than two brackets. While this can seem confusing at first, by working step by step in sequence, it becomes much easier and manageable for students. Students who have already learn the technique can try their hands on these questions.
Example 1:
Expand \((4x-3)(2x+5)\)
Solution:
\(
\left(4x-3\right)\left(2x+5\right)\)
\(=\left(4x\right)\left(2x\right)+\left(4x\right)\left(5\right)\)
\(+\left(-3\right)\left(2x\right)+\left(-3\right)\left(5\right)\)
\(=8x^2+14x-15\)
Example 2:
Expand\(5x(4x-3)(2x+5)\)
Solution:
Example 2 is an extension of Example 1, so we will make use of the solution in Example 1. What we will do is we will expand the last 2 brackets, and then multiple with the first term \(5x\).
\(5x\left(4x-3\right)\left(2x+5\right)\)
\(=5x\left(8x^2+14x-15\right)\)
\(=40x^3+70x^2-75x\)
Expansion For Perfect Square/Cube
Next, we introduce perfect squares and perfect cubes, teaching students how to recognize and work with these special patterns.
Perfect squares are numbers that, when square root, the result is an integer. For example:
1, 4, 9, 16, 25, 36, 49, …
Perfect cubes are numbers that, when cube root, the result is an integer. For example:
1, 8, 27, 64, 125, …
Students already familiar with Perfect Square Identities can skip this video. We will be covering the techniques to expand perfect square/cube with 2 examples:
Example 1:
Expand \((3x+5)^2\)
Solution:
\(\left(3x+5\right)^2\)
\(=\left(3x+5\right)\left(3x+5\right)\)
\(=9{x}^2+30x+25\)
Example 2:
Expand \((3x+5)^3\)
Solution:
\(\left(3x+5\right)^3\)
\(=\left(3x+5\right)\left(3x+5\right)\left(3x+5\right)\)
\(=\left(3x+5\right)\left(9{x}^2+30x+25\right)\)
\(=27{x}^3+135{x}^2+225x+125\)
Watch Full Concept Breakdown
Expansion of brackets is often clearer and more effective when explained through video rather than text alone. The video below walks through the step-by-step method for expanding brackets, using the examples listed above to demonstrate each stage clearly.