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Writing Expressions in Fully-Factored Form

(Factorising Expressions)

In this section we learn how to fully-factorise (or fully-factorize) expressions. We'll often be required to write an expression in its fully-factored form. We'll sometimes come across the term completely factored.

For instance, we may be asked to write the expression: \[6x^2 + 2x\] in its fully-factored forms. By the end of this section we'll have no trouble showing that: \[6x^2 + 2x = 2x\begin{pmatrix}3x + 1\end{pmatrix}\]

The "trick" for doing this is to find the highest common factor of each of the terms inside the original expression and to place it as a factor. This is best explained in the following tutorial.

Tutorial 1: writing expressions in factored form

In the following tutorial we learn a three-step method for factorizing expressions.

Exercise 1

Write each of the following expressions in fully-factored form:

  1. \(14x-6\)
  2. \(4p+14\)
  3. \(5a-25\)
  4. \(6x^2 + 10x\)
  5. \(3x^2-6x\)

Answers Without Working

We find:

  1. \(3x^2-6x = 3x\begin{pmatrix}x - 2 \end{pmatrix}\)
  2. \(12ab^2 + 4a = 4a\begin{pmatrix} 3b^2 + 1 \end{pmatrix}\)
  3. \(24p^2q - 8 p^3q^4 = 8p^2q \begin{pmatrix}3 - pq^3 \end{pmatrix}\)
  4. \(7m^3n^2 + 14n^3 = 7n^2 \begin{pmatrix} m^3 + 2n\end{pmatrix}\)
  5. \(15xyz^2 + 10 y^2z^3 = 5yz^2 \begin{pmatrix} 3x + 2yz \end{pmatrix}\)

Tutorial 2: Important Trick

A Must-Know Trick

At times one of the terms of the expression equals to the Highest Common Factor (HCF) of all the terms in the expression.
When this happens we'll need to use the special trick that's explained in this tutorial.
To illustrate this trick, we write the expression: \[6m^2-3m\] in its fully-factored form.

Exercise 2

Write each of the following in fully-factored form:

  1. \(6x-3\)
  2. \(15m + 3\)
  3. \(12b^2 + 4b\)
  4. \(12p^3-3p^2\)
  5. \(-4x^4-6x\)

Tutorial 3

Some more complicated examples

At times one of the terms of the expression is itself equal to the Highest Common Factor (HCF) of all the terms in the expression.
When this happens we'll need to use the special trick that's explained in this tutorial.
To illustrate this trick, we write the expression: \[6m^2-3m\] in its fully-factored form.

Exercise 3

  1. \(12ab^2 + 4a\)
  2. \(24p^2q - 8 p^3q^4\)
  3. \(7m^3n^2 + 14n^3\)
  4. \(15xyz^2 + 10 y^2z^3\)

Tutorial 4

Factoring 3 Terms or More

At times one of the terms of the expression is itself equal to the Highest Common Factor (HCF) of all the terms in the expression.
When this happens we'll need to use the special trick that's explained in this tutorial.
To illustrate this trick, we write the expression: \[6m^2-3m\] in its fully-factored form.

Exercise 4

Write each of the following expressions in fully-factored form:

  1. \(2x^2y - 4xy^3 + 8x\)
  2. \(7m^2n^2+21m^3n - 14m^5n^4\)
  3. \(10x^5 - 20y^3 + 30xz^3\)
  4. \(6a^4b^2 + 9a^2b^3-18a^5b^2\)
  5. \(9p^2q^3r + 18p^3q^5r^2 - 27pqr\)


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