Given the problem where we need to find a degree 4 polynomial with the specified zeros $-1, 0, 4, 1.4$ and a coefficient of $x^3$ equal to 13, let's proceed with the following steps:

Step 1: Form the Polynomial

Since the polynomial has zeros at $-1$, $0$, $4$, and $1.4$, we can express it as: $P(x) = k \cdot (x + 1)(x)(x - 4)(x - 1.4)$ where $k$ is a constant to be determined.

Step 2: Expand the Polynomial

First, we'll expand the factors:

1. Multiply the first two factors: $(x + 1)(x) = x^2 + x$.
2. Multiply the last two factors: $(x - 4)(x - 1.4) = x^2 - 5.4x + 5.6$.

Combining these results, we have: $P(x) = k \cdot (x^2 + x)(x^2 - 5.4x + 5.6)$

Now, let's expand this expression: $(x^2 + x)(x^2 - 5.4x + 5.6) = x^4 - 5.4x^3 + 5.6x^2 + x^3 - 5.4x^2 + 5.6x$

Combining like terms: $x^4 - 4.4x^3 + 0.2x^2 + 5.6x$

So, the polynomial becomes: $P(x) = k(x^4 - 4.4x^3 + 0.2x^2 + 5.6x)$

Step 3: Determine the Constant $k$

The coefficient of $x^3$ is given as 13, so we set up the equation: $-4.4k = 13$ $k = -\frac{13}{4.4} = -\frac{65}{22}$

Step 4: Write the Final Polynomial

Substituting $k = -\frac{65}{22}$ into the polynomial: $P(x) = -\frac{65}{22} \left( x^4 - 4.4x^3 + 0.2x^2 + 5.6x \right)$

Expanding this expression gives the final polynomial: $P(x) = -\frac{65}{22}x^4 + 65x^3 - \frac{13}{11}x^2 - 16.5x$

Final Answer:

The polynomial of degree 4 that satisfies the given conditions is: $P(x) = -\frac{65}{22}x^4 + 65x^3 - \frac{13}{11}x^2 - 16.5x$

Would you like further details or have any questions?

Here are some related questions to explore further:

1. What is the effect of different roots on the shape of a polynomial graph?
2. How can we determine the end behavior of polynomials of various degrees?
3. What are the properties of polynomials when all roots are real versus when some are complex?
4. How does the leading coefficient of a polynomial affect its graph?
5. Can the degree of a polynomial be determined if not all roots are known?

Tip: When expanding polynomials, carefully combine like terms to simplify and verify the degree and coefficients of the resulting expression.