What is the standard form of $y = (2 x^{2} + 2) (x + 5) {(x - 1)}^{2}$ $y= (2x^2 + 2)(x + 5)(x -1)^2$ ?

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What is the standard form of $y = (2 x^{2} + 2) (x + 5) {(x - 1)}^{2}$ $y= (2x^2 + 2)(x + 5)(x -1)^2$ ?

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Answer 1 · 2016-01-22T21:23:51

$2 x^{5} + 8 x^{4} - 16 x^{3} + 16 x^{2} - 18 x + 10$ $2x^5 + 8x^4 - 16x^3 +1 6x^2 - 18x + 10$

Explanation:

Expand the 2 'pairs' of brackets

ie $(2 x^{2} + 2) (x + 5) and (x - 1) (x - 1)$ $(2x^2 + 2)(x + 5 ) and (x - 1 )(x - 1 )$

using the FOIL method on each pair to obtain :

$(2 x^{3} + 10 x^{2} + 2 x + 10) (x^{2} - x - x + 1)$ $(2x^3 + 10x^2 + 2x + 10 )(x^2 - x - x + 1 )$

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

Now each term in the 2nd bracket must be multiplied by each
term in the 1st.

ie $2 x^{3} (x^{2} - 2 x + 1) + 10 x^{2} (x^{2} - 2 x + 1) + 2 x (x^{2} - 2 x + 1)$ $2x^3(x^2 -2x + 1) + 10x^2(x^2 - 2x + 1 ) + 2x(x^2 - 2x + 1 )$

$+ 10 (x^{2} - 2 x + 1)$ $+ 10(x^2 - 2x + 1 )$

$= 2 x^{5} - 2 x^{4} + 2 x^{3} + 10 x^{4} - 20 x^{3} + 10 x^{2} + 2 x^{3} - 4 x^{2}$ $= 2x^5 - 2x^4 + 2x^3 + 10x^4 - 20x^3 + 10x^2 + 2x^3 - 4x^2$

$+ 2 x + 10 x^{2} - 20 x + 10$ $+ 2x + 10x^2 - 20x + 10$

now collect 'like terms'

$= 2 x^{5} + 8 x^{4} - 16 x^{3} + 16 x^{2} - 18 x + 10$ $= 2x^5 + 8x^4 - 16x^3 + 16x^2 - 18x + 10$

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What is the standard form of $y = (2 x^{2} + 2) (x + 5) {(x - 1)}^{2}$ $y= (2x^2 + 2)(x + 5)(x -1)^2$ ?

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Explanation:

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What is the standard form of $y = (2 x^{2} + 2) (x + 5) {(x - 1)}^{2}$ $y= (2x^2 + 2)(x + 5)(x -1)^2$ ?