2. Unit 3: Algebraic Skills
  3. Lesson 3: Factoring — Difference of Squares and Perfect Squares

Factoring Difference of Squares


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Investigating a Pattern

Recall

Factoring and expanding are opposites.

Let's learn something about factoring by looking at three expanding questions:

 

 

 

Investigating a Pattern Continued

Expanding Question 3:

Consider the term \(9x^3y^4+2z^2\).

 

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Difference of Squares

  • A difference of squares is a subtraction of two terms, each of which is a perfect square.
  • A difference of squares can be factored using the rule  \(a^2-b^2=(a-b)(a+b)\).

Verification of the Rule

To verify the rule for factoring a difference of squares, we start with the right side and expand.

\(\begin{align*}(a-b)(a+b)& = a^2+ab-ab-b^2\\ &=a^2-b^2 \end{align*}\)

Thus, we have proven the rule.

Example 1

Factor fully, where possible.

  1. \(16x^2-9\)
  2. \(x^2+9\)

Solution — Part A

For \(16x^2-9\), notice the following:

  • The expression is a difference (a subtraction).
  • Each term is a perfect square since \(16x^2=(4x)^2\) and \(9=(3)^2\).
  • Therefore, \(16x^2-9\) is a difference of squares, \(a^2-b^2\), with \(a=4x\) and \(b=3\).
  • Therefore, \(16x^2-9 =(4x-3)(4x+3)\).

Solution — Part B

For \(x^2+9\), notice the following:

  • The expression is not a difference (it is a sum).
    • We cannot factor it as a difference of squares.
  • The expression is of the form \(x^2+bx+c\) with \(b=0\) and \(c=9\). We can try to factor by determining two integers that add to \(0\) and multiply to \(9\).
    • No such integers exist. (This will always be the case for a sum of squares).
  • Therefore, \( x^2+9\)  is not factorable. As it turns out, the corresponding parabola defined by \(y=x^2+9\) has no zeros (it does not cross the \(x\)-axis but lies entirely above the \(x\)-axis).

Check Your Understanding 1


Factor \( ((((a)*(D))*(i))*(s))*(p) x^2 - (((b)*(V))*(a))*(l)\) fully.

Enter the factored form with each factor in parentheses.  Example: "(x-2)(x+2)".

Factored form:  There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:

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Example 2

Factor fully, where possible.

  1. \(x^{12}-16\)
  2. \(50x^5y^8-2x\)

Solution — Part A

For \(x^{12}-16\), notice the following:

Notice that the second factor, \(x^6-4\), is also a difference of squares, this time with \(a=x^3\) and \(b=2\). Thus, the full solution to this question is:

\(\begin{align*}x^{12}-16 & =(x^6+4)(x^6-4)\\ & =(x^6+4)(x^3+2)(x^3-2) \end{align*}\)

Note:

  • The expression is a difference (a subtraction).
  • Each term is a perfect square since \(x^{12}=(x^{6})^2\) and \(16=4^2\).
  • Therefore, \(x^{12}-16\) is a difference of squares with \(a=x^6\) and \(b=4\).
  • So, \( x^{12}-16 =(x^6+4)(x^6-4)\).

Solution — Part B

For \(50x^5y^8-2x\), notice the following:

Removing the common factor:

\(50x^5y^8-2x =2x(25x^4y^8-1)\)

Notice that \(25x^4y^8-1\) is a difference of squares with \(a=5x^2y^4\) and \(b=1\). Therefore,

\(\begin{align*}50x^5y^8-2x& =2x(25x^4y^8-1)\\ & =2x(5x^2y^4-1)(5x^2y^4+1) \end{align*}\)

Note:

  • The expression is a difference (a subtraction).
  • The terms are not perfect squares.
    • Thus, difference of squares factoring cannot be used … yet.
  • There is a common factor of \(2x\).

Check Your Understanding 2


Factor \((((((exp(r))*(exp(a)))*(n))*(d))*(e))*(d)\) fully.

Hint: Make sure you factor fully. This question involves more than one step.

Enter \(x^b\) as \(x\)^\(b\). Enter factors in parentheses.  Example: "(x^2-1)(x+3)".

Factored form:  There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:

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