Consider the following. f(x)=x4+20x2−125 a) Write the polynomial as the product of factors that are irreducible over the rationals. f(x)=? b) Write the polynomial as the product of linear and quadratic factors that are irreducible over the reals. f(x)=? c) Write the polynomial in completely factored form. f(x)=?

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Consider the following.  f(x)=x4+20x2−125  a) Write the polynomial as the product of factors that are irreducible over the rationals.  f(x)=?  b) Write the polynomial as the product of linear and quadratic factors that are irreducible over the reals.  f(x)=?  c) Write the polynomial in completely factored form.  f(x)=?

unessodopunsep · Accepted Answer

Step 1  Given that   f(x)=x4+20x2−125  Here write the polynomial as the product of factors that are irreducible over the rationals.  Step 2  (a) f(x)=x4+20x2−125  =x4−5x2+25x2−125  =x2(x2−5)+25(x2−5)  =(x2−5)(x2+25)  Here both (x2−5) and (x2+25) are irreducible over the rationals.  Therefore,  f(x)=(x2−5)(x2+25)  Step 3  (b) f(x)=(x2−5)(x2+25)  =(x+5)(x−5)(x2+25)  Therefore  f(x)=(x+5)(x−5)(x2+25) is the product of irreducible factors over R.  Step 4   (d) f(x)=x4+20x2−125   =(x2−5)(x2+25)   =(x+5)(x−5)(x2+25)   =(x+5)(x−5)(x+5i)(x−5i)   Therefore   f(x)=(x+5)(x−5)(x+5i)(x−5i), which is in completely factored form.

Consider the following. f(x)=x^{4}+20x^{2}-125 a) Write the polynomial as the product of

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