Question

question
what is the factored form of this polynomial?
p(x) = x³ + 7x² + 4x − 12
select the correct answer from each drop-down menu. for help, see this
p(x) = ( )( )( )

Explanation:

Step1: Find a root using Rational Root Theorem

Possible rational roots are factors of 12 over factors of 1, so ±1, ±2, ±3, ±4, ±6, ±12. Test \(x = 1\): \(1 + 7 + 4 - 12 = 0\). So \(x - 1\) is not a factor. Test \(x = -1\): \(-1 + 7 - 4 - 12 = -10
eq 0\). Test \(x = 2\): \(8 + 28 + 8 - 12 = 32
eq 0\). Test \(x = -2\): \(-8 + 28 - 8 - 12 = 0\). So \(x = -2\) is a root, so \((x + 2)\) is a factor.

Step2: Perform polynomial division or use synthetic division

Divide \(p(x)\) by \((x + 2)\). Using synthetic division with root -2:
Coefficients: 1 (x³), 7 (x²), 4 (x), -12 (constant)
Bring down 1. Multiply by -2: -2. Add to 7: 5. Multiply by -2: -10. Add to 4: -6. Multiply by -2: 12. Add to -12: 0.
So the quotient is \(x² + 5x - 6\).

Step3: Factor the quadratic

Factor \(x² + 5x - 6\). Find two numbers that multiply to -6 and add to 5: 6 and -1. So \(x² + 5x - 6 = (x + 6)(x - 1)\).

Step4: Combine factors

Thus, \(p(x) = (x + 2)(x + 6)(x - 1)\).

Answer:

\(p(x) = (x + 2)(x + 6)(x - 1)\)