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655-705 (Hard)|   Algebra|      
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x^4 + 12x^3 + 49x^2 + 78x + 40 can be factored as a product of four 17 Sep 2018, 01:24
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C
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65% (hard)

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56% (02:02) correct 44% (02:06) wrong based on 245 sessions

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\(x^4+12x^3+49x^2+78x+40\) can be factored as a product of four linear polynomials with integer coefficients. Which of following cannot be a factor of \(x^4+12x^3+49x^2+78x+40?\)

A. \( x+1\)

B. \( x+2\)

C. \( x+3\)

D. \( x+4\)

E. \( x+5\)
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C
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x^4 + 12x^3 + 49x^2 + 78x + 40 can be factored as a product of four 19 Sep 2018, 02:12
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Assume \(x^4+px^3+qx^2+rx+s = (x+a)(x+b)(x+c)(x+d).\)
Then, \(s = abcd.\)
The constant terms \(a, b, c\) and \(d\) of the linear factors are factors of the constant term s of the original polynomial.

Since \(3\) is not a factor of \(40, x + 3\) cannot be a factor of the original polynomial \(x^4+12x^3+49x^2+78x+40.\)

Therefore, C is the answer.
Answer: C
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x^4 + 12x^3 + 49x^2 + 78x + 40 can be factored as a product of four 17 Sep 2018, 02:34
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Trying the options 1 by 1.

Option 1: If x+1 is a factor then x = -1 should satisfy the equation \(x^4\)+12\(x^3\)+49\(x^2\)+78x+40

Equation becomes zero. Hence x+1 is a factor of \(x^4\)+12\(x^3\)+49\(x^2\)+78x+40.

Similarly trying other options.

Only x = -3 doesn't satisfy.

We can conclude that x+3 is not factor of \(x^4\)+12\(x^3\)+49\(x^2\)+78x+40.

C is the answer.
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x^4 + 12x^3 + 49x^2 + 78x + 40 can be factored as a product of four 17 Sep 2018, 08:33
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[Math Revolution GMAT math practice question]

\(x^4+12x^3+49x^2+78x+40\) can be factored as a product of four linear polynomials with integer coefficients. Which of following cannot be a factor of \(x^4+12x^3+49x^2+78x+40?\)

\(A. x+1\)
\(B. x+2\)
\(C. x+3\)
\(D. x+4\)
\(E. x+5\)
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\(?\,\,\,:\,\,\,\,{x^4} + 12{x^3} + 49{x^2} + 78x + 40\,\,\, \ne \,\,\,\left( {{\text{altern}}{\text{.}}\,\,{\text{choice}}} \right) \cdot p\left( x \right)\)

\(\left( A \right)\,\,\,{x^4} + 12{x^3} + 49{x^2} + 78x + 40 = \left( {x + 1} \right) \cdot p\left( x \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\, - 1\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{root}}\,\,{\text{of}}\,\,{x^4} + 12{x^3} + 49{x^2} + 78x + 40 = 0\)
\({\left( { - 1} \right)^4} + 12{\left( { - 1} \right)^3} + 49{\left( { - 1} \right)^2} + 78\left( { - 1} \right) + 40\mathop = \limits^? 0\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( A \right)\,\,{\text{refuted}}\)

\(\left( B \right)\,\,\,{x^4} + 12{x^3} + 49{x^2} + 78x + 40 = \left( {x + 2} \right) \cdot p\left( x \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\, - 2\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{root}}\,\,{\text{of}}\,\,{x^4} + 12{x^3} + 49{x^2} + 78x + 40 = 0\)
\({\left( { - 2} \right)^4} + 12{\left( { - 2} \right)^3} + 49{\left( { - 2} \right)^2} + 78\left( { - 2} \right) + 40\mathop = \limits^? 0\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( B \right)\,\,{\text{refuted}}\)

\(\left( C \right)\,\,\,{x^4} + 12{x^3} + 49{x^2} + 78x + 40 = \left( {x + 3} \right) \cdot p\left( x \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\, - 3\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{root}}\,\,{\text{of}}\,\,{x^4} + 12{x^3} + 49{x^2} + 78x + 40 = 0\)
\({\left( { - 3} \right)^4} + 12{\left( { - 3} \right)^3} + 49{\left( { - 3} \right)^2} + 78\left( { - 3} \right) + 40\mathop = \limits^? 0\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( C \right)\,\,{\text{is}}\,\,{\text{the}}\,\,{\text{right}}\,\,{\text{answer}}\)


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

Regards,
Fabio.
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x^4 + 12x^3 + 49x^2 + 78x + 40 can be factored as a product of four 21 Sep 2023, 11:14
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Can this concept get tested on GMAT ?
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x^4 + 12x^3 + 49x^2 + 78x + 40 can be factored as a product of four 25 Sep 2023, 15:57
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Can this concept get tested on GMAT ?
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Hi, pramodmundhra15 !

Why not? All the definitions and properties envolved in the question stem and in (at least some of the potential) solutions are in the GMAT quant section scope.

Besides that, proper solutions take 2min or less to the implemented, therefore I cannot see any reason to doubt this question could be in the real exam, for sure!

Good luck and success in your studies,
Fabio.
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x^4 + 12x^3 + 49x^2 + 78x + 40 can be factored as a product of four 26 Sep 2023, 01:35
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fskilnik i believe GMAT doesn't test beyond quadratic equations.
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x^4 + 12x^3 + 49x^2 + 78x + 40 can be factored as a product of four 26 Sep 2023, 05:18
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fskilnik i believe GMAT doesn't test beyond quadratic equations.
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Hi again, pramodmundhra15 .

Polynomials are part of the GMAT scope even for degrees greater than 2, mainly in "factoring issues" like this one.

I will not continue this discussion any further, though.

Best Regards,
Fabio.
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x^4 + 12x^3 + 49x^2 + 78x + 40 can be factored as a product of four 08 Nov 2024, 07:39
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