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Factor the perfect square trinomial completely. z^6-10z^3+25 z^6-10z^3+25= square

Pergunta

Factor the perfect square trinomial completely.
z^6-10z^3+25
z^6-10z^3+25= square

Factor the perfect square trinomial completely. z^6-10z^3+25 z^6-10z^3+25= square

Solução

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CintiaMestre · Tutor por 5 anos

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To factor the given perfect square trinomial \( z^6 - 10z^3 + 25 \), let's follow these steps:<br /><br />### Step 1: Recognize the structure of a perfect square trinomial<br />A perfect square trinomial has the form:<br />\[<br />a^2 - 2ab + b^2 = (a - b)^2<br />\]<br /><br />Here, we need to identify \( a^2 \), \( -2ab \), and \( b^2 \) in the given expression.<br /><br />### Step 2: Rewrite the terms<br />The given trinomial is:<br />\[<br />z^6 - 10z^3 + 25<br />\]<br /><br />- The first term, \( z^6 \), can be written as \( (z^3)^2 \), so \( a = z^3 \).<br />- The last term, \( 25 \), can be written as \( 5^2 \), so \( b = 5 \).<br />- The middle term, \( -10z^3 \), matches \( -2ab \), where \( a = z^3 \) and \( b = 5 \), because:<br />\[<br />-2ab = -2(z^3)(5) = -10z^3<br />\]<br /><br />### Step 3: Write the factored form<br />Since the trinomial fits the form \( a^2 - 2ab + b^2 \), it factors as:<br />\[<br />(z^3 - 5)^2<br />\]<br /><br />### Final Answer:<br />\[<br />z^6 - 10z^3 + 25 = (z^3 - 5)^2<br />\]
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