This problem has two parts, Part A and Part B.
Part A: Use the Remainder Theorem to find the remainder when p(x)=x^3+4x^2+5x+2 is divided by x+1.
Part B: Then, use the Factor Theorem to determine whether x+1 is a factor of p(x)=x^3+4x^2+5x+2.
Select one answer for Part A and one answer for Part B.
A. The remainder is 0
A. The remainder is 2
A. The remainder is 12
B. No, x + 1 is not a factor of p(x)
B. Yes, x + 1 is a factor of p (x)
can someone help me with this question I'm confused I feel like the remainder is 2 and x + 1 is a factor of p(x) am I wrong or could somebody please help
When p(x)=x^3+4x^2+5x+2 is divided by x+1
the p(-1) will be the remainder, so
p(-1) = (-1)^3 + 4(-1)^2 + 5(-1) + 2
= -1 + 4 - 5 + 2
= 0
Therefore x+1 is a factor of p(x)
Thank you for explaining properly this really helped!
Sure! Let's break down the problem step by step.
Part A requires us to use the Remainder Theorem to find the remainder when dividing the polynomial p(x) by x+1.
To use the Remainder Theorem, we substitute -1 (the opposite sign of the divisor) into the polynomial p(x) and calculate the value.
So, plugging -1 into p(x) = x^3 + 4x^2 + 5x + 2, we get:
p(-1) = (-1)^3 + 4(-1)^2 + 5(-1) + 2
Simplifying this expression, we have:
p(-1) = -1 + 4 + (-5) + 2
p(-1) = 0
Therefore, the remainder is 0. Therefore, the correct answer for Part A is "The remainder is 0."
Now let's move on to Part B, which asks us to use the Factor Theorem to determine whether x+1 is a factor of p(x).
To check whether x+1 is a factor of p(x), we need to see if p(-1) is equal to 0 (since a polynomial has a factor of x - a if and only if it evaluates to 0 at that value).
From Part A, we found that p(-1) = 0, which means x+1 is a factor of p(x).
Therefore, the correct answer for Part B is "Yes, x + 1 is a factor of p(x)."
So, you were correct in stating that the remainder is 0 and x + 1 is a factor of p(x).