Examine the validity of the following statement: $(x+3)$ is a factor of $(x^{2}+10x+21)$.
(TRUE) To determine if $(x+3)$ is a factor of $p(x) = x^{2}+10x+21$,we use the Factor Theorem.
According to the Factor Theorem,$(x-a)$ is a factor of $p(x)$ if $p(a) = 0$.
Here,$(x+3) = (x - (-3))$,so $a = -3$.
Now,calculate $p(-3)$:
$p(-3) = (-3)^{2} + 10(-3) + 21$
$p(-3) = 9 - 30 + 21$
$p(-3) = 30 - 30 = 0$.
Since $p(-3) = 0$,the statement is True.
According to the Factor Theorem,$(x-a)$ is a factor of $p(x)$ if $p(a) = 0$.
Here,$(x+3) = (x - (-3))$,so $a = -3$.
Now,calculate $p(-3)$:
$p(-3) = (-3)^{2} + 10(-3) + 21$
$p(-3) = 9 - 30 + 21$
$p(-3) = 30 - 30 = 0$.
Since $p(-3) = 0$,the statement is True.
Explore More
Similar Questions
What must be subtracted from $8x^4 + 14x^3 - 2x^2 + 8x - 12$ so that the resulting polynomial is exactly divisible by $4x^2 + 3x - 2$?
Difficult
View SolutionAnswer the following and justify: Can $x^{2}-1$ be the quotient on division of $x^{6}+2 x^{3}+x-1$ by a polynomial in $x$ of degree $5$?
Medium
View SolutionExamine the validity of the statement: $(x+3)$ is a factor of $x^{3}+6x^{2}+5x-12$.
Easy
View SolutionThe zeros of cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$; $a \neq 0, a, b, c, d \in R$ are $\alpha, \beta$ and $\gamma$; then $\alpha\beta + \beta\gamma + \gamma\alpha = \ldots$
Medium
View SolutionObtain a quadratic polynomial $p(x) = ax^2 + bx + c$,where the sum of zeros is $\sqrt{2}$ and the product of zeros is $\frac{1}{3}$,given that $a < 0$.
Medium
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