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Use the remainder theorem and the factor theorem to determine whether (c + 5) is a factor of (c4 + 7c3 + 6c2 − 18c + 10)
A. The remainder isn't 0 and, therefore, (c + 5) is a factor of (c4 + 7c3 + 6c2 − 18c + 10)
B. The remainder is 0 and, therefore, (c + 5) isn't a factor of (c4 + 7c3 + 6c2 − 18c + 10)
C. The remainder is 0 and, therefore, (c + 5) is a factor of (c4 + 7c3 + 6c2 − 18c + 10)
D. The remainder isn't 0 and, therefore, (c + 5) isn't a factor of (c4 + 7c3 + 6c2 − 18c + 10)

Respuesta :

C. Use remainder theorem: If we divide a polynomial f(x) by (x-c) the remainder equals f(c). c=-5 in our case, so the remainder is f(-5). Plug in c=-5 into c4 + 7c3 + 6c2 − 18c + 10, f(-5)=0. Since the remainder is 0, (c+5) is a factor of the polynomial.
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Answer:

The correct answer is C.

Step-by-step explanation:

The given expression is

[tex]c^4+7c^3+6c^2-18c+10[/tex]

Let [tex]p(c)=c^4+7c^3+6c^2-18c+10[/tex].


According to the Remainder Theorem, if we divide a polynomial , [tex]p(c)[/tex] by [tex]c-a[/tex],then the remainder is [tex]p(a)[/tex].


The Factor Theorem is a special case of the remainder theorem, According to this theorem, if [tex]p(a)=0[/tex], then [tex]c-a[/tex] is a factor of [tex]p(c)[/tex].


We set [tex]c+5=0[/tex], this implies that, [tex]c=-5[/tex].


We substitute [tex]c=-5[/tex], to obtain,


Let [tex]p(-5)=(-5)^4+7(-5)^3+6(-5)^2-18(-5)+10[/tex].

We evaluate to obtain,

[tex]p(-5)=625+7(-125)+6(25)-18(-5)+10[/tex]


[tex]p(-5)=625-875+150+90+10[/tex].

We simplify to get,

[tex]p(-5)=-10+10[/tex]


[tex]\Rightarrow p(-5)=0[/tex]

The remainder is [tex]0[/tex] and therefore [tex](c+5)[/tex] is a factor of [tex]c^4+7c^3+6c^2-18c+10[/tex].

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