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4 October, 00:44

How do I factor 125x^3+8

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  1. 4 October, 02:11
    0
    Step 1:

    53x3 - 8

    Step 2:

    Trying to factor as a Difference of Cubes:

    2.1 Factoring: 125x3-8

    Theory : A difference of two perfect cubes, a3 - b3 can be factored into

    (a-b) • (a2 + ab + b2)

    Proof : (a-b) • (a2+ab+b2) =

    a3+a2b+ab2-ba2-b2a-b3 =

    a3 + (a2b-ba2) + (ab2-b2a) - b3 =

    a3+0+0+b3 =

    a3+b3

    Check : 125 is the cube of 5

    Check : 8 is the cube of 2

    Check : x3 is the cube of x1

    Factorization is:

    (5x - 2) • (25x2 + 10x + 4)

    Trying to factor by splitting the middle term

    2.2 Factoring 25x2 + 10x + 4

    The first term is, 25x2 its coefficient is 25.

    The middle term is, + 10x its coefficient is 10.

    The last term, "the constant", is + 4

    Step-1 : Multiply the coefficient of the first term by the constant 25 • 4 = 100

    Step-2 : Find two factors of 100 whose sum equals the coefficient of the middle term, which is 10.

    -100 + - 1 = - 101

    -50 + - 2 = - 52

    -25 + - 4 = - 29

    -20 + - 5 = - 25

    -10 + - 10 = - 20

    -5 + - 20 = - 25

    For tidiness, printing of 12 lines which failed to find two such factors, was suppressed

    Observation : No two such factors can be found!

    Conclusion : Trinomial can not be factored

    Final result:

    (5x - 2) • (25x2 + 10x + 4)
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