Question

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)

Answer 1

The correct answer is C.

Explanation:

The given expression is

`c^4+7c^3+6c^2-18c+10`

Let
`p(c)=c^4+7c^3+6c^2-18c+10`.

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

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

We set
`c+5=0`, this implies that,
`c=-5`.

We substitute
`c=-5`, to obtain,

Let
`p(-5)=(-5)^4+7(-5)^3+6(-5)^2-18(-5)+10`.

We evaluate to obtain,

`p(-5)=625+7(-125)+6(25)-18(-5)+10`

`p(-5)=625-875+150+90+10`.

We simplify to get,

`p(-5)=-10+10`

`\Rightarrow p(-5)=0`

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

Answer 2

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.