មេរៀន៖ កន្សោមពីជគណិត - ថ្នាក់ទី៩ ភាគ ៥ (Algebraic Expressions - Grade 9 parts-5)

មេរៀន៖ កន្សោមពីជគណិត - ថ្នាក់ទី៩ (Algebraic Expressions - Grade 9)
(Full Step-by-Step Answer Key)

ភាគ ៥៖ វិធីគុណខ្វែង (ញែកតួកណ្តាល) (Part 5: Cross-Multiplication Method)

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៣០. ដាក់កន្សោមជាផលគុណកត្តាតាមវិធីគុណខ្វែង (ញែកតួកណ្តាល)

ក. $3x^2+20x-7$

$x$	✘	$+7$	$\longrightarrow$	$+21x$
$3x$		$-1$	$\longrightarrow$	$-x$
$+20x$

$= (x+7)(3x-1)$

ខ. $2x^2-9x-5$

$x$	✘	$-5$	$\longrightarrow$	$-10x$
$2x$		$+1$	$\longrightarrow$	$+x$
$-9x$

$= (x-5)(2x+1)$

គ. $6a^2+13a+6$

$2a$	✘	$+3$	$\longrightarrow$	$+9a$
$3a$		$+2$	$\longrightarrow$	$+4a$
$+13a$

$= (2a+3)(3a+2)$

ឃ. $6x^2+7x-10$

$x$	✘	$+2$	$\longrightarrow$	$+12x$
$6x$		$-5$	$\longrightarrow$	$-5x$
$+7x$

$= (x+2)(6x-5)$

ង. $4a^2+11a+6$

$a$	✘	$+2$	$\longrightarrow$	$+8a$
$4a$		$+3$	$\longrightarrow$	$+3a$
$+11a$

$= (a+2)(4a+3)$

ច. $4x^2-25x+6$

$x$	✘	$-6$	$\longrightarrow$	$-24x$
$4x$		$-1$	$\longrightarrow$	$-x$
$-25x$

$= (x-6)(4x-1)$

ឆ. $14x^2+15x-9$

$2x$	✘	$+3$	$\longrightarrow$	$+21x$
$7x$		$-3$	$\longrightarrow$	$-6x$
$+15x$

$= (2x+3)(7x-3)$

ជ. $24y^2-35y+4$

$3y$	✘	$-4$	$\longrightarrow$	$-32y$
$8y$		$-1$	$\longrightarrow$	$-3y$
$-35y$

$= (3y-4)(8y-1)$

ឈ. $10y^2-23y+12$

$2y$	✘	$-3$	$\longrightarrow$	$-15y$
$5y$		$-4$	$\longrightarrow$	$-8y$
$-23y$

$= (2y-3)(5y-4)$

ញ. $9x^2+18x+8$

$3x$	✘	$+4$	$\longrightarrow$	$+12x$
$3x$		$+2$	$\longrightarrow$	$+6x$
$+18x$

$= (3x+4)(3x+2)$

ដ. $27x^2-6x-8$

$3x$	✘	$-2$	$\longrightarrow$	$-18x$
$9x$		$+4$	$\longrightarrow$	$+12x$
$-6x$

$= (3x-2)(9x+4)$

ឋ. $24x^2+17x-20$

$3x$	✘	$+4$	$\longrightarrow$	$+32x$
$8x$		$-5$	$\longrightarrow$	$-15x$
$+17x$

$= (3x+4)(8x-5)$

៣១. ដាក់កន្សោមជាផលគុណកត្តា (បន្ថែមនិងបន្ថយតួ ឬគុណខ្វែង)

ក. $x^2+5x+4$

$= x^2 + 4x + x + 4$
$= x(x+4) + 1(x+4)$
$= (x+4)(x+1)$

ខ. $x^2+5x+6$

$= x^2 + 3x + 2x + 6$
$= x(x+3) + 2(x+3)$
$= (x+3)(x+2)$

គ. $t^2+8t+15$

$= t^2 + 5t + 3t + 15$
$= t(t+5) + 3(t+5)$
$= (t+5)(t+3)$

ឃ. $x^2-10x+9$

$= x^2 - 9x - x + 9$
$= x(x-9) - 1(x-9)$
$= (x-9)(x-1)$

ង. $t^2-11t+28$

$= t^2 - 7t - 4t + 28$
$= t(t-7) - 4(t-7)$
$= (t-7)(t-4)$

ច. $x^2+7x-8$

$= x^2 + 8x - x - 8$
$= x(x+8) - 1(x+8)$
$= (x+8)(x-1)$

ឆ. $x^2+x-6$

$= x^2 + 3x - 2x - 6$
$= x(x+3) - 2(x+3)$
$= (x+3)(x-2)$

ជ. $x^2+11x-12$

$= x^2 + 12x - x - 12$
$= x(x+12) - 1(x+12)$
$= (x+12)(x-1)$

ឈ. $b^2+6b-7$

$= b^2 + 7b - b - 7$
$= b(b+7) - 1(b+7)$
$= (b+7)(b-1)$

ញ. $x^2+3x-4$

$= x^2 + 4x - x - 4$
$= x(x+4) - 1(x+4)$
$= (x+4)(x-1)$

ដ. $y^2-y-12$

$= y^2 - 4y + 3y - 12$
$= y(y-4) + 3(y-4)$
$= (y-4)(y+3)$

ឋ. $y^2-2y-35$

$= y^2 - 7y + 5y - 35$
$= y(y-7) + 5(y-7)$
$= (y-7)(y+5)$

ឌ. $n^2-4n-12$

$= n^2 - 6n + 2n - 12$
$= n(n-6) + 2(n-6)$
$= (n-6)(n+2)$

ឍ. $a^2-3a-18$

$= a^2 - 6a + 3a - 18$
$= a(a-6) + 3(a-6)$
$= (a-6)(a+3)$

ណ. $x^2-6x-7$

$= x^2 - 7x + x - 7$
$= x(x-7) + 1(x-7)$
$= (x-7)(x+1)$

ត. $5t^2+12t+7$

$= 5t^2 + 5t + 7t + 7$
$= 5t(t+1) + 7(t+1)$
$= (t+1)(5t+7)$

ថ. $2x^2+13x-7$

$= 2x^2 + 14x - x - 7$
$= 2x(x+7) - 1(x+7)$
$= (x+7)(2x-1)$

ទ. $2x^2+5x-3$

$= 2x^2 + 6x - x - 3$
$= 2x(x+3) - 1(x+3)$
$= (x+3)(2x-1)$

៣២. ចាប់កត្តារួមមុន និងញែកតួកណ្តាល

ក. $2a^2+24a+70$

$= 2(a^2+12a+35)$
$= 2(a^2+7a+5a+35)$
$= 2[a(a+7) + 5(a+7)]$
$= 2(a+7)(a+5)$

ខ. $2x^2-4x-160$

$= 2(x^2-2x-80)$
$= 2(x^2-10x+8x-80)$
$= 2[x(x-10) + 8(x-10)]$
$= 2(x-10)(x+8)$

៣៣. ចាប់កត្តារួមមុន និងញែកតួកណ្តាលបន្ត

ក. $5x^2+15x+10$

$= 5(x^2+3x+2)$
$= 5(x^2+2x+x+2)$
$= 5[x(x+2) + 1(x+2)]$
$= 5(x+2)(x+1)$

ខ. $5x^2-15x+10$

$= 5(x^2-3x+2)$
$= 5(x^2-2x-x+2)$
$= 5[x(x-2) - 1(x-2)]$
$= 5(x-2)(x-1)$

គ. $x^3+5x^2-14x$

$= x(x^2+5x-14)$
$= x(x^2+7x-2x-14)$
$= x[x(x+7) - 2(x+7)]$
$= x(x+7)(x-2)$

ឃ. $2x^2-2x-112$

$= 2(x^2-x-56)$
$= 2(x^2-8x+7x-56)$
$= 2[x(x-8) + 7(x-8)]$
$= 2(x-8)(x+7)$

៣៤. ដាក់កន្សោមខាងក្រោមជាផលគុណកត្តា

ក. $8x^2+12x-36$

$= 4(2x^2+3x-9)$
$= 4(2x^2+6x-3x-9)$
$= 4[2x(x+3) - 3(x+3)]$
$= 4(x+3)(2x-3)$

ខ. $45x^2-6x-24$

$= 3(15x^2-2x-8)$
$= 3(15x^2-12x+10x-8)$
$= 3[3x(5x-4) + 2(5x-4)]$
$= 3(5x-4)(3x+2)$

៣៥. ដាក់កន្សោមខាងក្រោមជាផលគុណកត្តា

ក. $(x+1)^2-x-1$

$= (x+1)^2 - (x+1)$
$= (x+1)[(x+1) - 1]$
$= (x+1)(x)$ ឬ $= x(x+1)$

ខ. $(x+1)^2+3x+3$

$= (x+1)^2 + 3(x+1)$
$= (x+1)[(x+1) + 3]$
$= (x+1)(x+4)$

គ. $3(x-1)^2-x+1$

$= 3(x-1)^2 - (x-1)$
$= (x-1)[3(x-1) - 1]$
$= (x-1)(3x-3-1)$
$= (x-1)(3x-4)$

ឃ. $(a+b)^2-3a-3b$

$= (a+b)^2 - 3(a+b)$
$= (a+b)[(a+b) - 3]$
$= (a+b)(a+b-3)$

៣៦. កំណត់តម្លៃ a

ក. $4x^2-3ay^4 = (2x+9y^2)(2x-9y^2)$

យើងមាន $(2x+9y^2)(2x-9y^2) = (2x)^2 - (9y^2)^2$
$= 4x^2 - 81y^4$

ផ្ទឹមសមភាព៖ $4x^2 - 3ay^4 = 4x^2 - 81y^4$
ទាញបាន៖ $-3a = -81$
នាំឲ្យ $a = 27$

ខ. $16x^2+5ay^4 = (4x+5y^2)(4x-5y^2)$

យើងមាន $(4x+5y^2)(4x-5y^2) = (4x)^2 - (5y^2)^2$
$= 16x^2 - 25y^4$

ផ្ទឹមសមភាព៖ $16x^2 + 5ay^4 = 16x^2 - 25y^4$
ទាញបាន៖ $5a = -25$
នាំឲ្យ $a = -5$

៣៧. ដាក់កន្សោមខាងក្រោមជាផលគុណកត្តា

ក. $(x+1)^2-x^2+1$

$= (x+1)^2 - (x^2-1)$
$= (x+1)^2 - (x-1)(x+1)$
$= (x+1)[(x+1) - (x-1)]$
$= (x+1)(x+1-x+1)$
$= 2(x+1)$

ខ. $(x-1)^2+x^2-1$

$= (x-1)^2 + (x^2-1)$
$= (x-1)^2 + (x-1)(x+1)$
$= (x-1)[(x-1) + (x+1)]$
$= (x-1)(x-1+x+1)$
$= 2x(x-1)$

គ. $x^2-1-(x+1)^2$

$= (x^2-1) - (x+1)^2$
$= (x-1)(x+1) - (x+1)^2$
$= (x+1)[(x-1) - (x+1)]$
$= (x+1)(x-1-x-1)$
$= -2(x+1)$

ឃ. $(a-b)^2-a^2+b^2$

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

៣៨. ដាក់កន្សោមខាងក្រោមជាផលគុណកត្តា

ក. $(x+2)^2-(x+2)(2x-3)$

$= (x+2)[(x+2) - (2x-3)]$
$= (x+2)(x+2-2x+3)$
$= (x+2)(5-x)$

ខ. $(x-3)(x+4)-(x-3)^2$

$= (x-3)[(x+4) - (x-3)]$
$= (x-3)(x+4-x+3)$
$= 7(x-3)$

គ. $2(x+1)(x-1)-3(x-1)^2$

$= (x-1)[2(x+1) - 3(x-1)]$
$= (x-1)(2x+2-3x+3)$
$= (x-1)(5-x)$

ឃ. $x(x-2)-(x-2)^2+x^2-4$

$= x(x-2) - (x-2)^2 + (x^2-4)$
$= x(x-2) - (x-2)^2 + (x-2)(x+2)$
$= (x-2)[x - (x-2) + (x+2)]$
$= (x-2)(x-x+2+x+2)$
$= (x-2)(x+4)$