SOLUTION: Factor a negative number or a GCF with a negative coefficient from the polynomial. -5x^3+x^9

Question 1015000: Factor a negative number or a GCF with a negative coefficient from the polynomial.
-5x^3+x^9
Found 2 solutions by Fombitz, Theo:
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
your expression is -5x^3 + x^9

since x^9 is equal to (x^3)^3, your expression becomes:

-5x^3 + (x^3)^3

you can now factor out a common (x^3) from the expression.

you will get:

-5x^3 + (x^3)^3 = x^3 * (-5 + (x^3)^2)

since -5 is equal to -1 * 5 and 1 * (x^3)^2 is equal to -1 * -1 * (x^3)^2, you can factor out a -1 to get:

x^3 * (-5 + (x^3)^2) is equal to:
x^3 * (-1 * 5 + -1 * -1 * (x^3)^2) which is equal to:
x^3 * (-1) * (5 + -1 * (x^3)^2) which is equal to:
x^3 * (-1) * (5 - (x^3)^2).
since (x^3)^2 is equal to x^6, your expression becomes equal to:
-x^3 * (5 - x^6)

your last expression became that way because:

-1 * x^3 is equal to -x^3
and:
(x^3)^2 is equal to x^(2*3) which is equal to x^6

x^9 became equal to (x^3)^3 because (x^3)^3 is equal to x^(3*3) which is equal to x^9.

so.....

you started with -5x^3 + x^9

and.....

you ended up with -x^3 * (5 - x^6)

you factored out a greatest common factor with a negative coefficient so the problem requirements have been satisfied.

you can confirm by applying the distributive law of multiplication to this final expression to get:

-x^3 * (5 - x^6) equals:
5 * (-x^3) - (x^6) * (-x^3) which is equal to:
-5x^3 + x^6 * x^3) which is equal to:
-5x^3 + x^(6+3) which is equal to:
-5x^3 + x^9 which looks exactly like your original expression.

an alternative way to check if you got it right is to give a random value to x and solve both the original expression and the final expression to see if they come up with the same answer.

the original expression is -5x^3 + x^9

the final expression is -x^3 * (5 - x^6)

when x = 17 (chosen at random), the original expression is equal to 1.185878519 * 10^11 and the final expression is equal to 1.185878519 * 10^11.

they're the same, so the final expression is equivalent to the original expression.

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