Factor analysis

Algebra Level 2

Factorize ( a 2 + 7 a ) 2 − 8 ( a 2 + 7 a ) − 180 (a^2+7a)^2- 8(a^2+7a)-180 .

( a − 2 ) ( a + 6 ) ( a + 11 ) ( a + 101 ) (a-2 )(a+6) (a+11) (a+101) ( a − 19 ) ( a − 20 ) ( a + 25 ) ( a + 55 ) (a-19)(a-20)(a+25)(a+55) ( a − 2 ) ( a + 2 ) ( a + 5 ) ( a + 9 ) (a-2) (a+2) (a+5) (a+9) ( a − 3 ) ( a + 1 ) ( a + 8 ) ( a + 9 ) (a-3) (a+1) (a+8) (a+9)

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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2 solutions

Munem Shahriar
Apr 3, 2017

Suppose, a²+7a=x

=x² - 8x - 180

here, 1× (-180)= -180

now, (-18) × 10 = -180; (-18) +10= -8

now, x² - 18x + 10x -180

=x(x - 18) + 10 (x -18)

=(x - 18) (x + 10)

=(a² + 7a - 18) (a² + 7a +10)

= (a² + 9a - 2a - 18) (a² + 5a + 2a + 10)

= {a(a + 9) - 2(a + 9)} {a(a + 5) + 2(a + 5)}

=(a + 9) (a - 2) (a + 5) (a + 2)

=(a - 2) (a + 2) (a + 5) (a + 9)

Ans.(a-2) (a+2) (a+5) (a+9)

5 Helpful 0 Interesting 0 Brilliant 0 Confused

f ( a ) = ( a 2 + 7 a ) 2 − 8 ( a 2 + 7 a ) − 180 Let x = a 2 + 7 a = x 2 − 8 x − 180 = ( x − 18 ) ( x + 10 ) = ( a 2 + 7 a − 18 ) ( a 2 + 7 a + 10 ) = ( a − 2 ) ( a + 9 ) ( a + 2 ) ( a + 5 ) = ( a − 2 ) ( a + 2 ) ( a + 5 ) ( a + 9 ) \begin{aligned} f(a) & = {\color{#3D99F6}(a^2+7a)}^2 - 8{\color{#3D99F6}(a^2+7a)} - 180 & \small \color{#3D99F6} \text{Let }x = a^2 + 7a \\ & = {\color{#3D99F6}x}^2 - 8{\color{#3D99F6}x} - 180 \\ & = ({\color{#3D99F6}x} - 18)({\color{#3D99F6}x} + 10) \\ & = ({\color{#3D99F6}a^2+7a} - 18)({\color{#3D99F6}a^2+7a} + 10) \\ & = (a-2)(a+9)(a+2)(a+5) \\ & = \boxed{(a-2)(a+2)(a+5)(a+9)} \end{aligned}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

What is f ( a ) f(a) ?

Munem Shahriar - 3 years, 11 months ago

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f ( a ) = ( a 2 + 7 a ) 2 − 8 ( a 2 + 7 a ) − 180 f(a) = {\color{#3D99F6}(a^2+7a)}^2 - 8{\color{#3D99F6}(a^2+7a)} - 180 . I just introduce it to shorten the left-hand-side of the equation for presentation purpose. I should have written "Let f ( a ) = ( a 2 + 7 a ) 2 − 8 ( a 2 + 7 a ) − 180 f(a) = {\color{#3D99F6}(a^2+7a)}^2 - 8{\color{#3D99F6}(a^2+7a)} - 180 ." but I wanted to save a line.

Chew-Seong Cheong - 3 years, 11 months ago

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