FACTORING IN ALGEBRA USING ALGEBRA FORMULAS, SOLVED EXAMPLES, EXERCISE PROBLEMS WITH ANSWERS

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

That knowledge is a prerequisite here.

Example 1 : Factoring In Algebra

Factorize 64a3 - 336a2 + 588a - 343

solution to Example 1 of Factoring In Algebra :

Let P = 64a3 - 336a2 + 588a - 343
We know
64 = 4 x 4 x 4 = 43;
336 = 7 x 48 = 7 x 3 x 16 = 3 x 42 x 7;
588 = 7 x 84 = 7 x 7 x 12 = 72 x 3 x 4 = 3 x 4 x 72;
343 = 7 x 49 = 7 x 7 x 7 = 73
∴ P = (4a)3 - 3(4a)2(7) + 3(4a)(7)2 - 73
This looks like
(first term)3 - 3(first term)2(second term)
+ 3(first term)(second term)2 - (second term)3
which is equal to
(first term - second term)3 [See Formula 7],
with (4a) in place of first term and 7 in place of second term.
Applying this Formula here, we get
P = (4a - 7)3.

Thus Algebra Factoring of 64a3 - 336a2 + 588a - 343by using Algebra Formulas gave the factors as (4a - 7)3. Ans.

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Example 2 : Factoring In Algebra

Solved Example 2 of Factoring In Algebra :

Factorize a6 + 54a3 + 729

solution to Example 2 of Factoring In Algebra :

Let P = a6 + 54a3 + 729
We know 729 = 9 x 81 = 9 x 3 x 27 = 27 x 27 = 272; 54 = 2 x 27;
Let t = a3; then a6 = (a3)2 = t2
∴ P = t2 + 2(27)t + (27)2
This looks like a2 + 2ab + b2 which is equal to (a + b)2 [ See Formula 1], with (t) in place of a and 27 in place of b
∴ P = (t + 27)2
(t + 27) = a3 + 33 = (a + 3){a2 - a(3) + 32} [See Formula 4]
= (a + 3)(a2 - 3a + 9)
∴ P = (t + 27)2 = {(a + 3)(a2 - 3a + 9)}2= (a + 3)2(a2 - 3a + 9)2.

Thus Algebra Factoring of a6 + 54a3 + 729by using Algebra Formulas gave the factors as (a + 3)2(a2 - 3a + 9)2. Ans.

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Example 3 : Factoring In Algebra

Solved Example 3 of Factoring In Algebra :

If x3 + y3 + z3 = 3xyz, show that either x + y + z = 0 or x = y = z

solution to Example 3 of Factoring In Algebra :

By data x3 + y3 + z3 = 3xyzx3 + y3 + z3 - 3xyz = 0
We have a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca) [See Formula 9]
Applying this here, we get
(x + y + z)(x2 + y2 + z2 - xy - yz - zx) = 0
⇒ either (x + y + z) = 0 or (x2 + y2 + z2 - xy - yz - zx) = 0
Let P = (x2 + y2 + z2 - xy - yz - zx)
Multiplying both sides with 2, we get
2P = (2x2 + 2y2 + 2z2 - 2xy - 2yz - 2zx)
Writng 2x2 as x2 + x2, 2y2 as y2 + y2 and 2z2 as z2 + z2 and regrouping the terms, we get
2P = {(x2 + y2 - 2xy) + (y2 + z2 - 2yz) + (z2 + x2 - 2zx)}
= {(x - y)2 + (y - z)2 + (z - x)2}
P = 0 ⇒ 2P = 2 x 0 = 0 ⇒ {(x - y)2 + (y - z)2 + (z - x)2} = 0
SUM OF SQUARES OF THREE TERMS IS ZERO ⇒ EACH TERM IS ZERO.
∴ (x - y) = 0, (y - z) = 0 and (z - x) = 0 ⇒ x = y, y = z and z = x
x = y = z
Thus, If x3 + y3 + z3 = 3xyz, then either x + y + z = 0 or x = y = z.

Thus, by Algebra Factoring of x3 + y3 + z3 - 3xyz,
by using Algebra Formulas, we have proved that
either x + y + z = 0 or x = y = z if x3 + y3 + z3 = 3xyz.

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Example 4 : Factoring In Algebra

Solved Example 4 of Factoring In Algebra :

Factorize (4x - 3y)3 + (y - 3x)3 + (2y - x)3

solution to Example 4 of Factoring In Algebra :

Let P = (4x - 3y)3 + (y - 3x)3 + (2y - x)3
Let A = (4x - 3y), B = (y - 3x) and C = (2y - x)
Then A + B + C = (4x - 3y) + (y - 3x) + (2y - x) = 4x - 4x + 3y - 3y = 0
We can prove If A + B + C = 0, A3 + B3 + C3 = 3ABC
For proof, see the Example 12 of Algebra Formulas.
∴ A + B + C = 0 ⇒ A3 + B3 + C3 = 3ABC
∴ P = (4x - 3y)3 + (y - 3x)3 + (2y - x)3= A3 + B3 + C3 and also A + B + C = 0
∴ P = 3ABC = 3(4x - 3y)(y - 3x)(2y - x).

Thus Algebra Factoring of (4x - 3y)3 + (y - 3x)3 + (2y - x)3by using Algebra Formulas gave the factors as 3(4x - 3y)(y - 3x)(2y - x). Ans.

Exercise : Factoring In Algebra

Solve the following problems
in Factoring In Algebra.

  1. Factorize 125x3 - 75x2 + 15x - 1
  2. Factorize 64x6 + 16x3 + 1
  3. Factorize 729a6 - 64b6
  4. Factorize 27x3 + 64y3 + 36xy -1
  5. Factorize (x - y)3 + (y - z)3 + (z - x)3
For Answers See at the bottom of the Page.

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Answers to Exercise : Factoring In Algebra

Answers to the problems of Factoring In Algebra :

  1. (5x - 1)3
  2. (2x + 1)2(4x2 - 2x + 1)2
  3. (3x + 4y - 1)(9x2 + 16y2 + 1 + 12xy - 4y -3x)
  4. 3(x - y)(y - z)(z - x)

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