1 Section
6 Lessons
Lifetime

Expand all sectionsCollapse all sections

Chapter Explanation
6

Factorization

OffOn

Curriculum

Square Based Identities in Factorization

4. Difference of Two Squares

$a^2 - b^2 = (a + b)(a - b)$

Ex 1:

$49y^2 - 36 = (7y)^2 - (6)^2 = \mathbf{(7y + 6)(7y - 6)}$

Ex 2:

$p^4 - 81 = (p^2 + 9)(p^2 - 9) = \mathbf{(p^2 + 9)(p - 3)(p + 3)}$

Ex 3 (Mixed):

$a^2 - 2ab + b^2 - c^2 = (a - b)^2 - c^2 = \mathbf{(a - b + c)(a - b - c)}$

5. Perfect Square Trinomials

These are the results of squaring a binomial sum or difference.

Sum Identity:

$a^2 + 2ab + b^2 = (a + b)^2$

Ex:

$x^2 + 8x + 16 = \mathbf{(x + 4)^2}$

Ex:

$25m^2 + 30m + 9 = \mathbf{(5m + 3)^2}$

Difference Identity:

$a^2 - 2ab + b^2 = (a - b)^2$

Ex: $4y^2 - 12y + 9 = \mathbf{(2y - 3)^2}$

6.Perfect Square Trinomial (Difference): The result of squaring a binomial difference.

$a^2 - 2ab + b^2 = (a - b)^2$

Ex 1:

Factorize $4y^2 - 12y + 9$

Sol:

$(2y)^2 - 2(2y)(3) + (3)^2 = \mathbf{(2y - 3)^2}$

Ex 2:

Factorize $a^2 - 2ab + b^2 - c^2$

Sol:

$(a - b)^2 - c^2 = [(a - b) + c][(a - b) - c] = \mathbf{(a - b + c)(a - b - c)}$

7. Trinomials with Constant Factors (Middle Term Splitting)

$x^2 + (a + b)x + ab = (x + a)(x + b)$

Ex 1:

$x^2 + 5x + 6 = \mathbf{(x + 3)(x + 2)}$

Ex 2:

$y^2 - 7y + 12 = \mathbf{(y - 4)(y - 3)}$

Ex 3 ( $ax^2 + bx + c$ ):

$3x^2 + 10x + 3 = \mathbf{(x + 3)(3x + 1)}$

Ex 4 (Advanced):

$6x^2 + 5x - 6 = \mathbf{(2x + 3)(3x - 2)}$

Ex 5 (Substitution):

$(2a - b)^2 + 2(2a - b) - 8 = \mathbf{(2a - b - 2)(2a - b + 4)}$

8. Square of a Trinomial Standard Identity

Factoring expressions with three variables ( $a, b, \text{ and } c$ ).

Standard Identity:

$a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a + b + c)^2$

Ex 1: $4a^2 + b^2 + c^2 + 4ab + 2bc + 4ac = \mathbf{(2a + b + c)^2}$

Ex 2: $9a^2 + 4b^2 + 16 + 12ab + 16b + 24a = \mathbf{(3a + 2b + 4)^2}$

9. Square of a Trinomial Variable Signs Identity

$a^2 + b^2 + c^2 + 2ab - 2bc - 2ca = (a + b - c)^2$

Usage: Used when the signs of the product terms involving one specific variable are negative.

Ex: Factorize $25x^2 + y^2 + 4z^2 - 10xy - 4yz + 20zx$

Sol:

$25x^2 + y^2 + 4z^2 - 10xy - 4yz + 20zx$

$\quad \quad = (5x)^2 + (-y)^2 + (2z)^2 + 2(5x)(-y) + 2(-y)(2z) + 2(2z)(5x)$

$\quad \quad = \mathbf{(5x - y + 2z)^2}$

Prev Previous Basic Methods & Linear Identities in Factorization

Next Cubic & Advanced Identities in Factorization Next

Search

Search

Modal title

Main Content