Harder Special Products And Factors-Chapter 6- Class 8- Advance Math

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Exercise- 6

1. Factorize

(i) (b+c)(c+a)(a+b)+abc

Solution:

(b+c)(c+a)(a+b)+abc

= (b+c){c(a+b)+a(a+b)}+abc

= (b+c)(ac+bc+ $\large a^{2}$ +ab)+abc

= (b+c)( $\large a^{2}$ +ab+ac+bc)+abc

= (b+c){ $\large a^{2}$ +a(b+c)+bc}+abc

= $\large a^{2}$ (b+c)+a(b+c)(b+c)+bc(b+c)+abc

= a(b+c){a+b+c}+bc{b+c+a}

= (a+b+c){a(b+c)+bc}

= (a+b+c)(ab+ac+bc)

(ii) a( $\large b^2$ - $\large c^2$ )+b( $\large c^2$ - $\large a^2$ )+c( $\large a^2$ - $\large b^2$ )

Solution:

a( $\large b^2$ - $\large c^2$ )+b( $\large c^2$ - $\large a^2$ )+c( $\large a^2$ - $\large b^2$ )

= a(b+c)(b-c)+b $\large c^2$ -b $\large a^2$ + $\large a^2$ c- $\large b^2$ c

= a(b+c)(b-c)+b $\large c^2$ - $\large b^2$ c-b $\large a^2$ + $\large a^2$ c

= a(b+c)(b-c)-bc(b-c)- $\large a^2$ (b-c)

= (b-c){a(b+c)-bc- $\large a^2$ }

= (b-c)(ab+ac-bc- $\large a^2$ )

= (b-c)<(- $\large a^2$ +ab+ac-bc)

= (b-c){-a(a-b)+c(a-b)}

= (b-c)(a-b)(c-a)

(iii) $\large y^2$ $\large z^2$ ( $\large y^2$ - $\large z^2$ )+ $\large z^2$ $\large x^2$ ( $\large z^2$ - $\large x^2$ )+ $\large x^2$ $\large y^2$ ( $\large x^2$ - $\large y^2$ )

Solution:

$\large y^2$ $\large z^2$ ( $\large y^2$ - $\large z^2$ )+ $\large z^2$ $\large x^2$ ( $\large z^2$ - $\large x^2$ )+ $\large x^2$ $\large y^2$ ( $\large x^2$ - $\large y^2$ )

$\large =y^2z^2(y+z)(y-z)+z^4x^2-z^2x^4+x^4y^2-x^2y^4$ $\large =y^2z^2(y+z)(y-z)-z^2x^4+x^4y^2+z^4x^2-x^2y^4$ $\large =y^2z^2(y+z)(y-z)+x^4(y^2-z^2)+x^2(z^4-y^4)$ $\large =y^2z^2(y+z)(y-z)+x^4(y^2-z^2)-x^2(y^4-z^4)$ $\large =y^2z^2(y+z)(y-z)+x^4(y+z)(y-z)-x^2(y^2+z^2)(y^2-z^2)$ $\large =(y+z)(y-z)\left \{ y^2z^2+x^4-x^2(y^2+z^2) \right \}$ $\large =(y+z)(y-z)\left \{ y^2z^2+x^4-x^2y^2-x^2z^2 \right \}$ $\large =(y+z)(y-z)\left \{ y^2z^2-x^2z^2-x^2y^2+x^4\right \}$ $\large =(y+z)(y-z)\left \{ z^2(y^2-x^2)-x^2(y^2-x^2)\right \}$ $\large =(y+z)(y-z)(y^2-x^2)(z^2-x^2)$

$\large =-(y+z)(y-z)(y+x)(y-x)(z+x)(z-x)$

$\large =(y+z)(y-z)(x+y)\left \{ -(x-y) \right \}(z+x)(z-x)$ $\large =-(y+z)(y-z)(x+y)(x-y)(z+x)(z-x)$

Harder Special Products And Factors-Chapter 6- Class 8- Advance Math

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