MathRevolution wrote:

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Math Revolution GMAT math practice question]

When \(f(x)=ax^2+bx+c (a≠0)\), is \(x+1\) a factor of \(f(x)?\)

\(1) f(1)=0\)

\(2) f(-1)=0\)

\(a \ne 0\)

\(a{x^2} + bx + c\,\,\mathop = \limits^? \,\,\left( {x + 1} \right) \cdot p\left( x \right)\,\,\,\,\,\, \Leftrightarrow \,\,\,\,a{\left( { - 1} \right)^2} + b\left( { - 1} \right) + c\,\,\mathop = \limits^? \,\,0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\boxed{b\,\,\mathop = \limits^? \,\,a + c}\)

\(\left( 1 \right)\,\,\,a + b + c = 0\,\,\,\,\,\left\{ \begin{gathered}

\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {2,0, - 2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\left( * \right)\, \hfill \\

\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {1,2, - 3} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\left( {**} \right)\,\, \hfill \\

\end{gathered} \right.\)

\(\left( * \right)\,\,\,2{x^2} - 2 = 2\left( {x + 1} \right)\left( {x - 1} \right) = \left( {x + 1} \right) \cdot p\left( x \right)\,\,\,,\,\,\,\,p\left( x \right) = 2\left( {x - 1} \right)\)

\(\left( {**} \right)\,\,{x^2} + 2x - 3 = \left( {x - 1} \right)\left( {x + 3} \right)\)

\(\left( 2 \right)\,\,a{\left( { - 1} \right)^2} + b\left( { - 1} \right) + c = 0\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle\)

The correct answer is therefore (B).

This solution follows the notations and rationale taught in the GMATH method.

Regards,

Fabio.

_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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