Question

select the correct answer
using synthetic division, what is the factored form of this polynomial?
x⁴ + 6x³ + 33x² + 150x + 200
a. (x + 2)(x + 4)(x² + 25)
b. (x + 2)(x + 4)(x - 5)(x + 5)
c. (x - 2)(x - 4)(x² + 25)
d. (x - 2)(x - 4)(x - 5)(x + 5)

Explanation:

Step1: Test possible roots

Possible rational roots are factors of 200 over factors of 1, so ±1, ±2, ±4, ±5, etc. Let's test \(x = -2\) using synthetic division.
Coefficients: 1 (x⁴), 6 (x³), 33 (x²), 150 (x), 200 (constant)
Bring down 1. Multiply by -2: -2. Add to 6: 4.
Multiply 4 by -2: -8. Add to 33: 25.
Multiply 25 by -2: -50. Add to 150: 100.
Multiply 100 by -2: -200. Add to 200: 0. So \(x + 2\) is a factor.
Now the quotient polynomial is \(x³ + 4x² + 25x + 100\).

Step2: Factor the cubic

Test \(x = -4\) on the cubic. Coefficients: 1, 4, 25, 100.
Bring down 1. Multiply by -4: -4. Add to 4: 0.
Multiply 0 by -4: 0. Add to 25: 25.
Multiply 25 by -4: -100. Add to 100: 0. So \(x + 4\) is a factor.
Quotient polynomial: \(x² + 25\).

Step3: Combine factors

Now we have factors \((x + 2)\), \((x + 4)\), and \(x² + 25\) (since \(x² + 25\) doesn't factor over reals). So the factored form is \((x + 2)(x + 4)(x² + 25)\).

Answer:

A. \((x + 2)(x + 4)(x² + 25)\)