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Factor the trinomial completely. 4q^2-11q-3 Select the correct choice below and, if necessary fill in the answer box within your choice. C A. 4q^2-11q-3=square B. The polynomial is prime

Pergunta

Factor the trinomial completely.
4q^2-11q-3
Select the correct choice below and, if necessary fill in the answer box within your choice.
C A.
4q^2-11q-3=square 
B. The polynomial is prime

Factor the trinomial completely. 4q^2-11q-3 Select the correct choice below and, if necessary fill in the answer box within your choice. C A. 4q^2-11q-3=square B. The polynomial is prime

Solução

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To factor the trinomial \(4q^2 - 11q - 3\), we need to find two binomials that multiply to give this expression. <br /><br />The general form for factoring a quadratic trinomial \(ax^2 + bx + c\) is to find two numbers that multiply to \(a \times c\) and add up to \(b\).<br /><br />1. Identify \(a\), \(b\), and \(c\):<br /> - \(a = 4\)<br /> - \(b = -11\)<br /> - \(c = -3\)<br /><br />2. Calculate \(a \times c = 4 \times (-3) = -12\).<br /><br />3. Find two numbers that multiply to \(-12\) and add up to \(-11\). These numbers are \(-12\) and \(1\).<br /><br />4. Rewrite the middle term \(-11q\) using \(-12\) and \(1\):<br /> \[<br /> 4q^2 - 12q + q - 3<br /> \]<br /><br />5. Factor by grouping:<br /> - Group the terms: \((4q^2 - 12q) + (q - 3)\)<br /> - Factor out the greatest common factor from each group:<br /> - From \(4q^2 - 12q\), factor out \(4q\): \(4q(q - 3)\)<br /> - From \(q - 3\), factor out \(1\): \(1(q - 3)\)<br /><br />6. Combine the factored groups:<br /> \[<br /> 4q(q - 3) + 1(q - 3) = (4q + 1)(q - 3)<br /> \]<br /><br />Thus, the completely factored form of the trinomial is:<br /><br />A. \(4q^2 - 11q - 3 = (4q + 1)(q - 3)\)
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