Question

question 2 (1 point) write the simplest polynomial in standard form with the given roots. -1 (multiplicity 2) and 4 \\(\circ x^3 + 2x^2 - 7x + 4\\) \\(\circ x^3 - 3x - 4\\) \\(\circ x^3 - 6x^2 - 7x + 4\\) \\(\circ x^3 - 2x^2 - 7x - 4\\)

Explanation:

Step1: Form factor from roots

For root $x=-1$ (multiplicity 2): $(x+1)^2$; for root $x=4$: $(x-4)$
Polynomial: $P(x)=(x+1)^2(x-4)$

Step2: Expand $(x+1)^2$

$(x+1)^2=x^2+2x+1$

Step3: Multiply by $(x-4)$

$P(x)=(x^2+2x+1)(x-4)=x^3-4x^2+2x^2-8x+x-4$

Step4: Combine like terms

$P(x)=x^3+(-4x^2+2x^2)+(-8x+x)-4=x^3-2x^2-7x-4$

Answer:

$\boldsymbol{x^3 - 2x^2 - 7x - 4}$ (matches the last option)