Ex1: Find an Equation of a Degree 4 Polynomial Function From the Graph of the Function

Mathispower4u

15 Jun 201204:04

Summary

TLDRThe video explains how to find the equation of a degree 4 polynomial in factored form using its x-intercepts and an additional point. The speaker identifies the polynomial's zeros at -3, -1, 2, and 5, and sets up the function with these factors. Then, by substituting the y-intercept (0, -15), the value of the constant 'A' is determined to be -1/2. The final polynomial function is presented, and the graph's behavior with a negative leading coefficient is discussed. This method demonstrates how to construct a polynomial from given roots and points.

Takeaways

🔢 The goal is to find an equation for a degree 4 polynomial function in factored form.
🔍 The function will have at most four real rational zeros, as indicated by its degree.
📉 The graph shows four rational zeros corresponding to the x-intercepts (-3, -1, +2, +5).
🧮 The polynomial can be written in factored form using the zeros (x + 3)(x + 1)(x - 2)(x - 5).
📐 A constant 'A' must be determined to complete the polynomial equation.
📝 To find 'A', the y-intercept (-15) is used, providing the point (0, -15) to substitute into the equation.
➗ By substituting x = 0 into the factored equation, the value of 'A' is found to be -1/2.
✅ The final polynomial equation is f(x) = -1/2(x + 3)(x + 1)(x - 2)(x - 5).
⬇️ The polynomial has a negative leading coefficient, which means the graph approaches negative infinity on both sides.
📊 The behavior of the graph matches what is expected for a degree 4 polynomial with a negative leading coefficient.

Q & A

What is the degree of the polynomial function discussed in the script?
-The degree of the polynomial function is 4.
How many real rational zeros or roots does the polynomial function have?
-The polynomial function has four real rational zeros or roots.
What are the x-intercepts of the polynomial function, according to the graph?
-The x-intercepts of the polynomial function are -3, -1, +2, and +5.
How do you write the polynomial function in factored form using the zeros?
-The polynomial function in factored form is written as F(x) = A(x + 3)(x + 1)(x - 2)(x - 5), where A is a constant.
What additional information is needed to determine the value of the constant 'A'?
-To determine the value of 'A', we need an additional point on the graph, such as the y-intercept.
What is the y-intercept of the function, and how is it used to find 'A'?
-The y-intercept is -15, and the point (0, -15) is used to substitute into the equation to solve for 'A'.
How is the value of 'A' calculated from the point (0, -15)?
-Substituting 0 for x into the equation gives 30A = -15. Solving for A gives A = -1/2.
What is the final equation of the polynomial function in factored form?
-The final equation of the polynomial function in factored form is F(x) = -1/2(x + 3)(x + 1)(x - 2)(x - 5).
What does the leading coefficient tell us about the end behavior of the polynomial function?
-Since the leading coefficient is negative and the degree is even, the function approaches negative infinity in both directions.
Why do the signs of the constants in the factors have the opposite sign of the zeros from the graph?
-The constants in the factors have the opposite sign of the zeros because when we solve for the roots, setting each factor to zero gives the corresponding zero.

Outlines

00:00

🧮 Understanding Degree 4 Polynomial Functions

This paragraph explains how to find an equation for a degree 4 polynomial function and emphasizes that the function will be in factored form. It mentions that a degree 4 polynomial can have up to four real rational zeros or roots, and by examining the graph, it is clear that all roots are rational, corresponding to the x-intercepts (-3, -1, 2, 5). These intercepts are also the roots of the polynomial function, allowing us to write the function in its factored form.

🔍 Determining the Polynomial Factors

Here, the paragraph explains how to write the polynomial function in factored form using the identified roots. For each root, a corresponding factor is derived: X + 3, X + 1, X - 2, and X - 5. It notes that the signs of the constants in the factors are the opposite of the signs of the roots. This ensures that when these factors are substituted into the function, one of them will be zero, making the product zero and validating the function's zeros as shown in the graph.

📊 Finding the Constant 'A' Using a Point

The paragraph describes the necessity of finding the constant 'A' in the polynomial function, which requires another point from the graph. The y-intercept (-15) is used for this purpose. By substituting the x-value of 0 and setting the function equal to -15, a system is established to solve for 'A'. The substitution and simplification steps are detailed to find that 'A' equals -1/2.

✍️ Writing the Polynomial Function

Using the derived value of 'A' (-1/2) and the previously found factors, the polynomial function is written in its complete factored form: F(x) = -1/2 * (x + 3) * (x + 1) * (x - 2) * (x - 5). The paragraph highlights that if this expression is expanded, the leading coefficient would be negative, and it confirms the expected behavior of the function: approaching negative infinity in both directions, consistent with an even-degree polynomial with a negative leading coefficient.

📚 Conclusion and Next Steps

The final paragraph wraps up the discussion by expressing the hope that the explanation was helpful. It mentions that another example will be reviewed in the next video, indicating a continuation of the learning process.

Mindmap

Keywords

💡Degree 4 Polynomial

A degree 4 polynomial is a polynomial function with the highest exponent of the variable being 4. In the video, the function being analyzed is a degree 4 polynomial, meaning it can have up to four real roots. The graph of the function reflects this, as it has four x-intercepts.

💡Real Rational Zeros

Real rational zeros are the points where the graph of the polynomial crosses the x-axis, and these points are rational numbers. In the video, the graph of the degree 4 polynomial has four real rational zeros, which correspond to the x-intercepts at -3, -1, 2, and 5.

💡X-intercepts

The x-intercepts are the points where the graph crosses the x-axis, indicating the values of x for which the polynomial equals zero. In this case, the video points out the x-intercepts as (-3, 0), (-1, 0), (2, 0), and (5, 0), which are also the roots of the polynomial.

💡Roots/Zeros

Roots or zeros are the values of x that make the polynomial function equal to zero. These are also referred to as the x-intercepts on the graph. In the video, the polynomial has four roots at -3, -1, 2, and 5, which are used to factor the polynomial.

💡Factored Form

Factored form is the expression of a polynomial as a product of its factors. In the video, the degree 4 polynomial is expressed in factored form as (x + 3)(x + 1)(x - 2)(x - 5). This representation directly reflects the zeros of the function.

💡Constant 'A'

'A' is a constant that represents the leading coefficient of the polynomial in its factored form. To find the value of 'A', the video uses the y-intercept of the graph (0, -15), solving for 'A' by substituting values into the factored polynomial.

💡Y-intercept

The y-intercept is the point where the graph of the polynomial crosses the y-axis. In the video, the y-intercept is given as (0, -15), and it is used to find the value of the constant 'A' by substituting it into the polynomial equation.

💡Polynomial Function

A polynomial function is an expression consisting of variables raised to whole number powers and multiplied by coefficients. In the video, the polynomial function under discussion has a degree of 4, meaning its highest power of x is 4, and it can have up to four roots.

💡Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. In the video, the leading coefficient is determined to be negative (-1/2), which affects the end behavior of the polynomial's graph, causing it to approach negative infinity as x moves towards positive and negative infinity.

💡End Behavior

End behavior describes how the graph of a function behaves as x approaches infinity or negative infinity. In the video, the end behavior of the degree 4 polynomial is discussed, with the function approaching negative infinity on both sides due to the negative leading coefficient and even degree.

Highlights

We want to find an equation for the degree 4 polynomial function and leave the function in factored form.

A degree 4 polynomial function can have at most four real rational zeros or roots.

From the graph, all roots or zeros are rational because it has four x-intercepts: (-3, -1), (+2, +5).

The x-intercepts are also the roots or zeros of the polynomial function.

We can write the polynomial function in factored form using the roots.

To fully determine the equation, we need to find one more point, and we use the y-intercept, which is -15 at the point (0, -15).

We will first find the factors of the polynomial function from the zeros, then use the point to find the value of 'A'.

For a zero of x = -3, the factor is (x + 3).

For a zero of x = -1, the factor is (x + 1).

For a zero of x = +2, the factor is (x - 2).

For a zero of x = +5, the factor is (x - 5).

To find the value of 'A', substitute the point (0, -15) into the factored equation.

Solving for 'A', we get A = -1/2.

The final polynomial function in factored form is f(x) = -1/2 * (x + 3) * (x + 1) * (x - 2) * (x - 5).

The polynomial is degree 4 with a negative leading coefficient, so as x moves left and right, the function approaches negative infinity, which matches the graph behavior.