The quotient of the polynomials [tex]2x^{3} +3x^{2} -5x-6[/tex] divided by (x+1) is [tex]2x^{2} -x-6[/tex].
Given two polynomials [tex]2x^{3}+3x^{2} -5x-6[/tex] and (x+1).
We have to determine whether [tex]2x^{3}+3x^{2} -5x-6[/tex] is divisible by (x+1).
Factors are the numbers which are multiplied to get the number whose factors are given. The factor theorem is a theorem which links factors and zeroes of a polynomial.
Polynomial is an expression of more than two algebraic terms may be in addition or subtraction.
We have to first find the factors of polynomial [tex]2x^{3}+3x^{2} -5x-6[/tex].
[tex]2x^{3}+3x^{2} -5x-6[/tex]=[tex]2x^{3}+3x^{2} -4x-x-6[/tex]
=[tex]x^{2}[/tex](2x+3)-2(2x+3)-x
=[tex](x^{2} -x-2)[/tex](2x+3)
=([tex]x^{2}[/tex]-x-2)(2x+3)
=[[tex]x^{2}[/tex]-2x+x-2](2x+3)
=[x(x-2)+1(x-2)](2x+3)
=(x+1)(x-2)(2x+3)
Because (x+1) is a factor of [tex]2x^{3}+3x^{2} -4x-x-6[/tex],so it is divisible by (x+1).
Hence using factor theorem we can say that [tex]2x^{3}+3x^{2} -4x-x-6[/tex]is divisible by (x+1).
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