Use the factor theorem to determine whether the binomial is a factor of the given polynomial.
1. t+sqrt2; P(t)=t^5+t^4+4t+4
2. z-i; P(z)=z^7+z^6+z^5+z^4+z^3+z^2+z+1
3. z+2i; P(z)=z^3+z^2+4z+4
Thanks
1. P(-sqrt2)=-4sqrt2+4-4sqrt2+4 No
2. P(i)=-i-1+i+1-i-1+i+1=0 Yes
3. P(-2i)=8i-4-8i+4=0 Yes
To determine whether a binomial is a factor of a given polynomial, we can use the factor theorem. The factor theorem states that if a polynomial P(x) can be divided by (x - a) without any remainder, then (x - a) is a factor of the polynomial.
Let's apply the factor theorem to each of the given problems:
1. For the first problem: t + √2, and P(t) = t^5 + t^4 + 4t + 4
To check if (t + √2) is a factor, we need to plug in -√2 into P(t) and see if the result is 0.
P(-√2) = (-√2)^5 + (-√2)^4 + 4(-√2) + 4
= -2√2 + 4 - 8√2 + 4
= -10√2 + 8
Since P(-√2) = -10√2 + 8 ≠ 0, (t + √2) is not a factor of P(t).
2. For the second problem: z - i, and P(z) = z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1
To check if (z - i) is a factor, we need to plug in i into P(z) and see if the result is 0.
P(i) = i^7 + i^6 + i^5 + i^4 + i^3 + i^2 + i + 1
= i + 1 - i - 1 + i - 1 + i + 1
= 0
Since P(i) = 0, (z - i) is a factor of P(z).
3. For the third problem: z + 2i, and P(z) = z^3 + z^2 + 4z + 4
To check if (z + 2i) is a factor, we need to plug in -2i into P(z) and see if the result is 0.
P(-2i) = (-2i)^3 + (-2i)^2 + 4(-2i) + 4
= -8i^3 + 4 - 8i^2 + 4
= -8(-i) + 4 - 8(-1) + 4
= 8i + 4 + 8 + 4
= 8i + 16
Since P(-2i) = 8i + 16 ≠ 0, (z + 2i) is not a factor of P(z).
To summarize:
1. (t + √2) is not a factor of P(t).
2. (z - i) is a factor of P(z).
3. (z + 2i) is not a factor of P(z).
I hope this explanation helps!