Question

Find the zeros of the function. Then sketch a graph of the function. \(g(x)=-2 x^5+2 x^4+40 x^3\)

Short answer

The zeros of the function are \(x= -5,0\). The graph of the function touches the x-axis at these zeros and curve back up because of the negative leading coefficient.

Step-by-step solution

- Factoring

The given function is \(g(x)=-2 x^5+2 x^4+40 x^3\). As we can see, we can factor out an \(x^3\) from all the terms to simplify it. We get something like \(x^3(-2x^2+2x+40)\)

- Finding the Zeros

To find the zeros of the function we set \(g(x)=0\). Hence we have \(-2x^2+2x+40=0\). Solving this quadratic equation by using the quadratic formula \(x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), it comes out that \(x= -5,0\)

- Plotting the Graph

With zeros \(x=-5,0\), plot these points onto the x-axis. With the function \(g(x)=-2 x^5+2 x^4+40 x^3\), the leading coefficient is negative, telling us that the graph ends in the third and first quadrants. Plot more points as required and join these points to make the graph. The graph will touch the x-axis at these zeros and curve back up.

Key concepts

Zeros of a Function

In mathematics, finding the zeros of a function is the same as finding the x-values for which the function equals zero. These are crucial because they tell us where the graph of the function will intersect the x-axis.
For a polynomial function like \(g(x) = -2x^5 + 2x^4 + 40x^3\), the zeros can be identified by setting the whole function equal to zero and solving for \(x\).
\[g(x) = 0\]
These x-values are called zeros, roots, or solutions of the equation. They might be integers, fractions, or even complex numbers depending on the degree and specific values of the polynomial. Understanding zeros helps sketch the function and predict its behavior.

Quadratic Formula

The quadratic formula is a crucial tool used to find the zeros of quadratic equations, which take the form \(ax^2 + bx + c = 0\).
The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, \(a\), \(b\), and \(c\) are numbers from the equation and \(x\) represents the unknown or variable you're solving for.

• \(b^2 - 4ac\) is called the discriminant.

The discriminant determines the nature of the roots:

• If it is positive, two real and distinct roots exist.
• If it is zero, only one real root exists (a repeated root).
• If it is negative, two complex roots exist.

In our exercise, the quadratic within the factored polynomial is solved using this formula. It provided zeros \(x = -5\) and \(x = 0\), showcasing how zeros can be directly obtained from solving polynomial equations.

Factoring Polynomials

Factoring polynomials is a method used to simplify polynomials and make equations easier to solve. When you factor, you are essentially rewriting a polynomial as a product of simpler polynomials or monomials.
For the function \(g(x) = -2x^5 + 2x^4 + 40x^3\), the process begins by identifying the greatest common factor. Here, \(x^3\) is common in all terms, so it can be factored out:
\(g(x) = x^3(-2x^2 + 2x + 40)\).
This factorization step is vital for simplifying the problem and finding zeros.
Once the polynomial is factored, solving the simpler equation \(-2x^2 + 2x + 40 = 0\), which can be addressed using the quadratic formula, becomes manageable.
Factoring helps break down complex problems into more digestible parts, leading to solutions like \(x = -5\) and \(x = 0\). This method streamlines the process of plotting or analyzing polynomial functions.