Max@Math Revolution wrote:[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 [spoiler]___(B)______[/spoiler] .
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
