Questions: Given the following polynomial function: p(x) = 6x^3 + x^2 + 3x + 8 According to the Rational Zero Theorem, the possible zeros for p(x) are ±(1,2,4,8)/(1,2,3,6) Substitution reveals that x = -1 is a zero of p(x). Perform synthetic division to express p(x) in the factored form given below. Find the values for a, b, c. p(x) = (x+1)(ax^2 + bx + c) a = b = c = Identify the nature of the remaining two zeros: complex, rational, irrational (type one):

Given the following polynomial function:
p(x) = 6x^3 + x^2 + 3x + 8

According to the Rational Zero Theorem, the possible zeros for p(x) are ±(1,2,4,8)/(1,2,3,6)
Substitution reveals that x = -1 is a zero of p(x). Perform synthetic division to express p(x) in the factored form given below. Find the values for a, b, c.
p(x) = (x+1)(ax^2 + bx + c)
a =

b =

c =

Identify the nature of the remaining two zeros: complex, rational, irrational (type one):

Transcript text: Given the following polynomial function: \[ p(x)=6 x^{3}+x^{2}+3 x+8 \] According to the Rational Zero Theorem, the possible zeros for $p(x)$ are $\pm\left(\frac{1,2,4,8}{1,2,3,6}\right)$ Substitution reveals that $x=-1$ is a zero of $p(x)$. Perform synthetic division to express $p(x)$ in the factored form given below. Find the values for $a, b, c$. \[ \begin{array}{l} p(x)=(x+1)\left(a x^{2}+b x+c\right) \\ a= \\ \square \end{array} \] \[ b= \] $\square$ \[ c= \] $\square$ Identify the nature of the remaining two zeros: complex, rational, irrational (type one): $\square$

Solution

Solution Steps

To solve this problem, we will use synthetic division to divide the polynomial $ p(x) = 6x^3 + x^2 + 3x + 8 $ by $ x + 1 $, since $ x = -1 $ is a known zero. This will help us express $ p(x) $ in the form $ (x+1)(ax^2 + bx + c) $. The coefficients $ a $, $ b $, and $ c $ will be determined from the result of the synthetic division. Finally, we will analyze the quadratic $ ax^2 + bx + c $ to determine the nature of its zeros.

Step 1: Perform Synthetic Division

We start with the polynomial $ p(x) = 6x^3 + x^2 + 3x + 8 $ and divide it by $ x + 1 $ using synthetic division. The coefficients of the polynomial are $ [6, 1, 3, 8] $ and the known zero is $ -1 $.

After performing synthetic division, we find the quotient to be $ 6x^2 - 5x + 3 $ and the remainder is $ 0 $. Thus, we can express $ p(x) $ as: \[ p(x) = (x + 1)(6x^2 - 5x + 3) \]

Step 2: Identify Coefficients

From the quotient $ 6x^2 - 5x + 3 $, we identify the coefficients: \[ a = 6, \quad b = -5, \quad c = 3 \]

Step 3: Determine the Nature of the Remaining Zeros

Next, we analyze the quadratic $ 6x^2 - 5x + 3 $ to determine the nature of its zeros. We calculate the discriminant $ D $ using the formula: \[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 6 \cdot 3 = 25 - 72 = -47 \] Since the discriminant $ D < 0 $, the quadratic has two complex zeros.

Final Answer

The values of $ a $, $ b $, and $ c $ are: \[ \boxed{a = 6}, \quad \boxed{b = -5}, \quad \boxed{c = 3} \] The nature of the remaining two zeros is: \[ \boxed{\text{complex}} \]

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