Question

1. use the remainder theorem to determine which of the following are factors of the polynomial ( p(x) = x^3 - 4x^2 - 47x + 210 ). select all that apply.

( square ) ( (x + 7) )
( square ) ( (x - 6) )
( square ) ( (x + 4) )
( square ) ( (x - 5) )

Explanation:

Step 1: Recall the Remainder Theorem

The Remainder Theorem states that if we divide a polynomial \( P(x) \) by \( (x - a) \), the remainder is \( P(a) \). If \( (x - a) \) is a factor of \( P(x) \), then the remainder \( P(a) = 0 \). For a factor of the form \( (x + b) \), we can rewrite it as \( (x - (-b)) \), so we need to evaluate \( P(-b) \) and check if it equals 0.

The polynomial is \( P(x)=x^{3}-4x^{2}-47x + 210 \)

Step 2: Check \( (x + 7) \)

Rewrite \( (x + 7) \) as \( (x-(-7)) \). We need to find \( P(-7) \)

$$
LATEXBLOCK0
$$

Since \( P(-7) = 0 \), \( (x + 7) \) is a factor.

Step 3: Check \( (x - 6) \)

We need to find \( P(6) \)

$$
LATEXBLOCK1
$$

Since \( P(6)=0 \), \( (x - 6) \) is a factor.

Step 4: Check \( (x + 4) \)

Rewrite \( (x + 4) \) as \( (x-(-4)) \). We need to find \( P(-4) \)

$$
LATEXBLOCK2
$$

Since \( P(-4)
eq0 \), \( (x + 4) \) is not a factor.

Step 5: Check \( (x - 5) \)

We need to find \( P(5) \)

$$
LATEXBLOCK3
$$

Since \( P(5) = 0 \), \( (x - 5) \) is a factor.

Answer:

(x + 7), (x - 6), (x - 5)