GATE CS 2016 Set 1 Q.9 (GA Q.9): Quantitative Aptitude

If $f(x) = 2x^7 + 3x - 5$ , which of the following is a factor of $f(x)$ ?

$(x^3 + 8)$

$(x - 1)$

$(2x - 5)$

$(x + 1)$

Official answer and solution

Solution

Use the Factor Theorem: $(x - a)$ is a factor of $f(x)$ if and only if $f(a) = 0$ .

We are given the polynomial:

f(x) = 2x^7 + 3x - 5

Test each candidate linear factor by substituting its root into $f(x)$ :

f(1) = 2(1)^7 + 3(1) - 5 = 2 + 3 - 5 = 0

Since $f(1) = 0$ , by the Factor Theorem, $(x - 1)$ is a factor of $f(x)$ .

For completeness, test other options:

For $(x + 1)$ : $f(-1) = 2(-1)^7 + 3(-1) - 5 = -2 - 3 - 5 = -10 \ne 0$
For $(2x - 5)$ : $f(2.5) = 2(2.5)^7 + 3(2.5) - 5 > 0$
For $(x^3 + 8)$ : root is $x = -2$ , where $f(-2) = 2(-128) - 6 - 5 = -267 \ne 0$

Therefore, $(x - 1)$ is a factor of $f(x)$ .

Final answer

Common trap

Testing $x = -1$ instead of $x = 1$ for the factor $(x - 1)$ .

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