Respuesta :
The remainder that will be obtained on dividing the considered polynomial p(x) by (x+2) is given by: Option A: -34
What are the factors of a polynomial?
If a polynomial is written in terms of smaller degree polynomial, all in multiplication with each other, then such smaller degree polynomials are called factors of the polynomial.
A polynomial is itself one of its factors.
1 is also a factor of every polynomials.
For this case, we're given that:
[tex]p(x) = x^3 - 4x^2 + ax + 20[/tex]
One of its factor is given as (x+1)
Thus, we have:
[tex]p(x) = (x+1)g(x)[/tex]
where g(x) is also a polynomial since when we divide a polynomial by its valid factor, the result is also a polynomial.
Now, as p(x) is 3 degere polynomial, and p(x) is 1 degree, thus, g(x) is a 2 degree polynomial (powers of x will add up to make the three degree polynomial).
The general form of a 2 degree polynomial is: [tex]rx^2 + sx + ct[/tex] for r, s and t as constants.
Let we have:
[tex]g(x) = rx^2 + sx + t[/tex]
Then we get:
[tex]p(x) = (x+1)(rx^2 + sx + t)\\x^3 - 4x^2+ax + 20 = rx^3 + (r+s)x^2 + (s+t)x + t[/tex]
Comparing coefficients, we get:
[tex]r = 1\\-4 = r+s\\a = s +t\\20 = t[/tex]
That gives us:
[tex]r+s=-4\\1+s = -4\\s = -5[/tex]
and
[tex]a = s+t\\a = -5 + 20 = 15[/tex]
Thus, the polynomial is [tex]p(x) = x^3 - 4x^2 + 15x + 20[/tex]
Dividing this polynomial by (x+2) by using long division method, we get the quotient as [tex]x^2 - 6x + 27[/tex] and remainder as -34
Thus, the remainder that will be obtained on dividing the considered polynomial p(x) by (x+2) is given by: Option A: -34
Learn more about long division of polynomials here:
https://brainly.com/question/14624349
