Answer:
d=0, c=-8
Step-by-step explanation:
We have that [tex]f(x)= x^4+cx+d[/tex]. We want f is divisible by [tex](x+2)^2[/tex]. This means that
[tex]\frac{f(x)}{(x+2)^2}=q(x)[/tex] or [tex] f(x) = (x+2)^2q(x)[/tex]
where q is a polynomial of degree less than 4 (since f is a polynomial of degree 4). In this case, since f is of degree 4 and (x+2)^2 is of degree 2, we have that q(x) is of degree 2(this is because when we multiply polynomials they degree adds up).
Then, q(x) is of the form [tex](ex^2+fx+g)[/tex].
We can expand the right hand side so
[tex]x^4+cx+d = (x^2+2x+4)(ex^2+fx+g)= ex^4+(f+2e)x^3+(g+2f+4e)x^2+(2g+4f)x+4g [/tex]. Since both polynomials are equal,the coefficients of each power of x, on both sides, must be equal.Then, we have the equations
[tex] e=1, (f+2e)=0, g+2f+4e=g+2(f+2e)=0, 2g+4f=c, 4g = d[/tex]
From the first three equations we get that e=1, f=-2, g=0. Then, from the last two equations we get that d=0 and c=-8.
We can check that the polynomial
[tex]f(x) = x^4-8x = (x+2)^2(x-2)(x)[/tex]
Answer:
c=32, d=48
Step-by-step explanation:
(Ax²+Bx+C)(x+2)²= x⁴+cx+d
Ax⁴+Bx³+Cx²+4Ax³+4Bx²+4Cx+4Ax²+4Bx+4C= x⁴+cx+d
Equating coeeficients on both sides
A=1
B+4A= 0
B=-4
C+4B+4A=0
C=12
4C+4B=c
c=32
4C=d
d=48
