Use the Factor Theorem to determine whether $g(x)$ is a factor of $p(x)$ in each of the following cases: $p(x) = x^3 + 3x^2 + 3x + 1$,$g(x) = x + 2$.
(D) According to the Factor Theorem,$g(x) = x - a$ is a factor of $p(x)$ if and only if $p(a) = 0$.
Here,$g(x) = x + 2$,so we set $x + 2 = 0$,which gives $x = -2$.
Now,we evaluate $p(-2)$:
$p(-2) = (-2)^3 + 3(-2)^2 + 3(-2) + 1$
$p(-2) = -8 + 3(4) - 6 + 1$
$p(-2) = -8 + 12 - 6 + 1$
$p(-2) = 4 - 6 + 1 = -1$
Since $p(-2) \neq 0$,by the Factor Theorem,$g(x)$ is not a factor of $p(x)$.
Here,$g(x) = x + 2$,so we set $x + 2 = 0$,which gives $x = -2$.
Now,we evaluate $p(-2)$:
$p(-2) = (-2)^3 + 3(-2)^2 + 3(-2) + 1$
$p(-2) = -8 + 3(4) - 6 + 1$
$p(-2) = -8 + 12 - 6 + 1$
$p(-2) = 4 - 6 + 1 = -1$
Since $p(-2) \neq 0$,by the Factor Theorem,$g(x)$ is not a factor of $p(x)$.
Explore More
Similar Questions
Verify whether the following is a zero of the polynomial indicated against it:
$p(x) = lx + m, \, x = -\frac{m}{l}$
$p(x) = lx + m, \, x = -\frac{m}{l}$
Easy
View SolutionClassify the following as linear,quadratic,and cubic polynomials:
$(i)$ $x^{2}+x$
$(ii)$ $x-x^{3}$
$(iii)$ $y+y^{2}+4$
$(i)$ $x^{2}+x$
$(ii)$ $x-x^{3}$
$(iii)$ $y+y^{2}+4$
Easy
View SolutionWrite $(3a + 4b + 5c)^2$ in expanded form.
Medium
View SolutionFactorise: $x^{3}-2x^{2}-x+2$
Medium
View SolutionWrite the following cube in expanded form: $(2x + 1)^3$
Easy
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo