Questions: Find a polynomial of the specified degree that satisfies the given conditions. Degree 4; zeros -1,0,3, 1/3; coefficient of x^3 is 7 f(x)=

Transcript text: Find a polynomial of the specified degree that satisfies the given conditions. Degree $4 ; \quad$ zeros $-1,0,3, \frac{1}{3} ; \quad$ coefficient of $x^{3}$ is 7 \[ f(x)= \]

Solution

Solution Steps

To find a polynomial of degree 4 with specified zeros and a given coefficient for $x^3$, we start by constructing the polynomial from its roots. The polynomial can be expressed as $f(x) = a(x + 1)(x)(x - 3)(x - \frac{1}{3})$. We then expand this expression and adjust the leading coefficient $a$ such that the coefficient of $x^3$ is 7.

Step 1: Construct the Polynomial

Given the zeros $ -1, 0, 3, \frac{1}{3} $, we can express the polynomial as: \[ f(x) = a(x + 1)(x)(x - 3)(x - \frac{1}{3}) \]

Step 2: Expand the Polynomial

Expanding the polynomial yields: \[ f(x) = x^4 - 2.3333x^3 - 2.3333x^2 + 1.0x \]

Step 3: Determine the Leading Coefficient

The coefficient of $ x^3 $ in the expanded polynomial is $ -2.3333 $. To ensure the coefficient of $ x^3 $ is $ 7 $, we set: \[ a = \frac{7}{-2.3333} = -3.0 \]

Step 4: Write the Final Polynomial

Substituting $ a $ back into the polynomial gives: \[ f(x) = -3.0x^4 + 7.0x^3 + 7.0x^2 - 3.0x \]