The value of c so that (x + 1) is a factor of the polynomial p(x) = 5x^4 + 7x^3– 2x^2– 3x + c is equal to 1
Solution:
Given polynomial is as follows
[tex]\mathrm{p}(x)=5 x^{4}+7 x^{3}-2 x^{2}-3 x+\mathrm{c}[/tex]
Also given that (x+1) is factor of given polynomial p(x).
Need to determine value of constant c.
We will be solving above problem using factor theorem.
Factor theorem says that if ( x – a) is a factor of any polynomial f(x) , then f(a) = 0.
As in our case f(x) = p(x) and factor of p(x) is (x + 1) , which can be rewritten as (x – ( -1)) , so "a" in our case is -1 .
According to factor theorem p(-1) = 0
On substituting x = -1 , in given expression of p(x) we get
[tex]\begin{array}{l}{\mathrm{p}(-1)=5(-1)^{4}+7(-1)^{3}-2(-1)^{2}-3(-1)+\mathrm{c}} \\\\ {\mathrm{p}(-1)=5-7-2+3+\mathrm{c}} \\\\ {\mathrm{p}(-1)=\mathrm{c}-1} \\\\ {\text { As } \mathrm{p}(-1)=0, \mathrm{c}-1=0} \\\\ {=>\mathrm{c}=1}\end{array}[/tex]
Hence value of c in the given polynomial is 1
