Question

1. for the polynomial function \\( p(x)=x^5 + x^4 - 11x^3 - 9x^2 + kx \\), two of the zeros are +1 and -2. it can be determined that the largest zero of this function, to the nearest tenth, is. (record your answer in the numerical - response section on the answer sheet.)

Explanation:

Step1: Use root x=1 to find k

Since $x=1$ is a zero, substitute into $P(x)$:
$P(1)=(1)^5+(1)^4-11(1)^3-9(1)^2+k(1)=0$
$1+1-11-9+k=0$
$-18+k=0$
$k=18$

Step2: Rewrite polynomial with k=18

$P(x)=x^5+x^4-11x^3-9x^2+18x$
Factor out $x$ first:
$P(x)=x(x^4+x^3-11x^2-9x+18)$

Step3: Factor quartic with known roots

Since $x=1$ and $x=-2$ are zeros, $(x-1)$ and $(x+2)$ are factors. Multiply them:
$(x-1)(x+2)=x^2+x-2$
Divide the quartic $x^4+x^3-11x^2-9x+18$ by $x^2+x-2$:
Using polynomial long division or synthetic division:
$\frac{x^4+x^3-11x^2-9x+18}{x^2+x-2}=x^2-9$

Step4: Factor the quadratic

$x^2-9=(x-3)(x+3)$

Step5: List all zeros

Combine all factors:
$P(x)=x(x-1)(x+2)(x-3)(x+3)$
Zeros are $x=0, 1, -2, 3, -3$

Step6: Identify largest zero

Sort zeros: $-3, -2, 0, 1, 3$
Largest zero is 3.0

Answer:

3.0