After Quadratics, it’s time to Introduce the Third Power

As learners become more confident with quadratic expansion, the next step is extending this to cubic expressions.

Once they are comfortable expanding two brackets, the natural next step is to introduce a third.

For example:

$(x+1)(x+1)$ gives $x^2+2x+1$ 

Extending this further:

$(x+1)(x+1)(x+1)$ expands to $x^3+3x^2+3x+1$.

The first and last terms follow that familiar pattern: they come from multiplying the first terms together and the constants together.

Increasing complexity can stall progress

Even when learners are confident with expanding two brackets, adding a third can feel like a sudden step up.

This increase in complexity often comes just as they are beginning to feel secure, which can make it frustrating for them (and for you).

As the process becomes longer, it’s often easy to lose track of terms or make small errors.

Noticing the change in structure

Expanding three sets is still based on what you did with binomials.

Now, however, they are starting to multiply a binomial by a trinomial.

Results will have potentially four terms, the $x^3$ term, the $x^2$ term, the $x$term and the constant number.

For emerging learners

For learners who are building their procedural knowledge, focus on what they know to support taking new steps.

Take the expression:

$(x+3)(x+4)(x+5)$

Multiply each term in the first set of brackets by each term in the second.

This leaves:

$(x^2+8x+15)(x+4)$

Now expand again, carefully multiplying each term and you are left with an expression with six terms:

$x^3+4x^2+8x^2+32x+15x+60$.

After combining like terms, this gives:

 $x^3+12x^2+47x+60$.

When learners see these repeatedly they will start to notice patterns.

This is where fluency develops.

For learners needing extension

Reviewing answers should become a natural step in maths practice.

Cubic expansion reveals patterns to allow for quick checks with completed work.

Recall the way to check that two binomial expressions have been properly multiplied.

Cubic expansion has two elements that follow the same pattern:

• The first and last terms in the polynomial are the product of the first and second terms, by themselves, in the binomials.
• The two middle terms are not as readily apparent, but they are there, and they always are.

There is a useful pattern that appears in this work, first identified by Blaise Pascal.

This pattern is often referred to as Pascal’s Triangle:

1

1  1

1  2  1

1  3  3  1

1  4  6  4  1

The numbers in each row relate directly to the coefficients that appear when expanding expressions like $(x+1)^3$.

This Week, Focus on:

• Consolidating the expansion process
• Looking for patterns to check accuracy

Checking that the work makes sense is crucial for building new connections.

Taking time to verify answers helps learners develop confidence.

There is no rush with this because expanding brackets is such a fundamental part of the National 5 pathway.

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