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Please study
Formulas required for Factoring Expressions
if you have not already done so.

That knowledge is a prerequisite here.

Example 1 :

Factorize 16x2y2 + 24xyz + 9z2

solution to Example 1 of Factoring Special Products :

Let P = 16x2y2 + 24xyz + 9z2
= (4xy)2 + 2(4xy)(3z) + (3z)2

This is like a2 + 2ab + b2
with (4xy) in place of a and 3z in place of b

We have a2 + 2ab + b2 = (a + b)2 [ See Formula 1]

∴ P = (4xy + 3z)2

Thus Algebra Factoring of 16x2y2 + 24xyz + 9z2
by using Algebra Formulas gave the factors as
(4xy + 3z)2. Ans.

Thus the problem on Factoring Expressions is solved.

Example 2 of Factoring Expressions :

Factorize (2a + 3b)2 - 2(2a + 3b)(a - b) + (a - b)2

solution to Example 2 of Factoring Special Products :

Let P = (2a + 3b)2 - 2(2a + 3b)(a - b) + (a - b)2
= (first term)2 - 2(first term)(second term) + (second term)2
= (first term - second term)2 [See Formula 2]

= {(2a + 3b) - (a - b)}2 = {2a + 3b - a + b}2
= {a + 4b}2

Thus Algebra Factoring of (2a + 3b)2 - 2(2a + 3b)(a - b) + (a - b)2
by using Algebra Formulas gave the factors as
{a + 4b}2 Ans.

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Example 3 of Factoring Expressions :

Factorize 16x4 + 4x2 + 1

solution to Example 3 of Factoring Special Products :

Let P = 16x4 + 4x2 + 1= (4x2)2 + 12 + (4x2)(1)
This looks like a2 + b2 + ab with (4x2) in place of a and 1 in place of b
To make it (a + b)2, we have to add one more ab { = (4x2)(1)} [See Formula 1]
Not to change the value, let us add and subtract.
Adding and subtracting (4x2)(1) to the R.H.S. of P, we get
P = (4x2)2 + 12 + (4x2)(1) + (4x2)(1) - (4x2)(1)
= (4x2)2 + 12 + 2(4x2)(1) - (4x2)(1)
= (4x2 + 1)2 - (2x)2
This is difference of squares of two terms which is equal to the product of the sum and difference of the two terms [See Formula 3].
∴ P = {(4x2 + 1) + (2x)}{(4x2 + 1) - (2x)}
= (4x2 + 2x + 1)(4x2 - 2x + 1).

Thus Algebra Factoring of 16x4 + 4x2 + 1 by using Algebra Formulas gave the factors as (4x2 + 2x + 1)(4x2 - 2x + 1). Ans.

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Example 4 of Factoring Expressions :

Factorize 27x3 + 27x2 + 9x + 1

solution to Example 4 of Factoring Special Products :

Let P = 27x3 + 27x2 + 9x + 1= (3x)3 + 3(3x)2(1) + 3(3x)(1)2 + (1)3
This looks like a3 + 3a2b + 3ab2 + b3
with (3x) in place of a and 1 in place of b.
We have a3 + 3a2b + 3ab2 + b3 = a3 + 3ab(a + b) + b3 = (a + b)3 [See Formula 6]
Applying this Formula here, we get
P = (3x + 1)3.

Thus Algebra Factoring of 27x3 + 27x2 + 9x + 1 by using Algebra Formulas gave the factors as (3x + 1)3. Ans.

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Exercise : Factoring Expressions

Do the following problems on Factoring Expressions.

  1. Factorize (ab + cd)2 + 18(ab + cd) + 81
  2. Factorize 49x2 - 28xyz + 4y2z2
  3. Factorize 625a4 + 25a2b2 + b4
  4. Factorize 8a3 + 60a2 + 150a + 125
For Answers to these problems on Factoring
Expressions See at the bottom of the Page.

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Answers to Exercise : Factoring Expressions

Answers to problems on Factoring Expressions :

  1. (ab + cd + 9)2
  2. (7x - 2yz)2
  3. (25a2 + b2 + 5ab)(25a2 + b2 - 5ab)
  4. (2a + 5)3

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