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elena-14-01-66 [18.8K]
2 years ago
11

Is (x-1) a factor of x^100+4x^99+3?Can you please show steps

Mathematics
1 answer:
V125BC [204]2 years ago
7 0

Answer:

The answer is no because (1)^{100}+4(1)^{99}+3 \neq 0.

Step-by-step explanation:

If (x-1) is a factor of x^{100}+4x^{99}+3, then (1)^{100}+4(1)^{99}+3 \text{ will be } 0.

Let's check.

(1)^{100}+4(1)^{99}+3

1+4(1)+3

1+4+3

5+3

8

Since 8 \neq 0, then (x-1) is not a factor of x^{100}+4x^{99}+3.

======================================================

So anyways more on how this works:

If \frac{P(x)}{x-c}=Q(x)+\frac{R}{x-c} where R is the remainder and Q(x) is the quotient of the division, then multiplying both sides by (x-c) gives P(x)=Q(x)(x-c)+R.

We see that x-c is a factor if the remainder is 0. (Just like we know 2 is a factor of 6 because when we divide 6 by 2 we get a remainder of 0.)

Anyways if we evaluate P(x) for x=c we get:

P(c)=Q(x)(c-c)+R

P(c)=Q(x)(0)+R

P(c)=R

So evaluating P for x=c will tell us the remainder of the quotient \frac{P(x)}{x-c} without actually performing division.

The remainder will tell us if x-c is a factor of P

If the remainder is 0, then the answer is yes.

If the remainder is not 0, then the answer is no.

(The question to be answer "Is this a factor of this?")

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