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Factor completely, use any method (MUST SHOW ALL WORK!!)x^4+6x^2−7

Sagot :

we have the expression

x^4+6x^2−7

Note that, if you substitute for x=1

(1)^4+6(1)^2-7

1+6-7

7-7=0

that means

x=1 is a root of the given equation

(x-1) is a factor of the given equation

so

Divide x^4+6x^2−7 by (x-1)

x^4+6x^2−7 : (x-1)

x^3+x^2+7x+7

-x^4+x^3

---------------

x^3+6x^2-7

-x^3+x^2

---------------

7x^2-7

-7x^2+7x

-------------

7x-7

-7x+7

----------

0

therefore

x^4+6x^2−7=(x-1)(x^3+x^2+7x+7)

Note that in the cubic function, if you substitute for x=-1

(-1)^3+(-1)^2+7(-1)+7=0

so

x=-1 is a root of the cubic function

(x+1) is a factor

Divide x^3+x^2+7x+7 by (x+1)

x^3+x^2+7x+7 : (x+1)

x^2+7

-x^3-x^2

-------------------

7x+7

-7x-7

----------

0

so

x^3+x^2+7x+7=x^2+7

therefore

x^4+6x^2−7=(x+1)(x-1)(x^2+7)

the answer is

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

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