Formula For Polynomial P(x) Of Degree 4 Explained

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In this article, we delve into the fascinating world of polynomials, specifically focusing on a polynomial P(x) of degree 4. We'll explore how to construct the formula for P(x) given specific information about its roots and a point it passes through. This problem combines several key concepts in algebra, including the relationship between roots and factors of a polynomial, the concept of multiplicity of roots, and how to determine the leading coefficient using a given point. Understanding these concepts is crucial for solving a wide range of polynomial problems and gaining a deeper appreciation for the structure and behavior of these mathematical objects.

We are given a polynomial P(x) of degree 4. This means the highest power of x in the polynomial is 4. We have the following information:

  • P(x) has a root of multiplicity 2 at x = 3. This indicates that the factor (x - 3) appears twice in the factored form of P(x).
  • P(x) has a root of multiplicity 1 at x = 0. This means (x - 0), or simply x, is a factor of P(x).
  • P(x) has a root of multiplicity 1 at x = -4. This implies that (x + 4) is a factor of P(x).
  • P(x) passes through the point (5, 36). This gives us a specific x and y value that satisfies the equation P(x) = y, which we can use to determine the leading coefficient.

Our goal is to find a formula for P(x) that satisfies all these conditions.

Before diving into the solution, let's clarify some key concepts:

  • Roots of a polynomial: The roots of a polynomial P(x) are the values of x for which P(x) = 0. These are also known as the zeros of the polynomial.
  • Multiplicity of a root: The multiplicity of a root refers to the number of times a particular root appears as a solution of the polynomial equation. For example, if a root r has a multiplicity of 2, it means the factor (x - r) appears twice in the factored form of the polynomial.
  • Factored form of a polynomial: A polynomial can be expressed in factored form as a product of linear factors, each corresponding to a root of the polynomial. The general form is P(x) = a(x - r1)(x - r2)...(x - rn), where a is the leading coefficient and r1, r2, ..., rn are the roots.
  • Degree of a polynomial: The degree of a polynomial is the highest power of the variable x in the polynomial. A polynomial of degree 4 has the general form P(x) = ax^4 + bx^3 + cx^2 + dx + e, where a is not zero.

Given the roots and their multiplicities, we can start constructing the formula for P(x). Since P(x) has a root of multiplicity 2 at x = 3, the factor (x - 3) appears twice. This gives us (x - 3)^2. P(x) also has roots of multiplicity 1 at x = 0 and x = -4, contributing the factors x and (x + 4), respectively. Therefore, we can express P(x) in the following form:

P(x) = a * x * (x + 4) * (x - 3)^2

Here, a is the leading coefficient, which we need to determine using the given point (5, 36).

We know that P(5) = 36. We can substitute x = 5 into the equation and solve for a:

36 = a * 5 * (5 + 4) * (5 - 3)^2

Simplify the equation:

36 = a * 5 * 9 * (2)^2

36 = a * 5 * 9 * 4

36 = 180a

Now, solve for a:

a = 36 / 180

a = 1 / 5

So, the leading coefficient a is 1/5.

Now that we have the leading coefficient, we can write the complete formula for P(x):

P(x) = (1/5) * x * (x + 4) * (x - 3)^2

This is the formula for the polynomial of degree 4 that satisfies the given conditions. We can expand this expression to obtain the polynomial in standard form, but the factored form is often more useful for analyzing the roots and behavior of the polynomial.

To expand the polynomial, we first expand the squared term:

(x - 3)^2 = x^2 - 6x + 9

Now, substitute this back into the formula for P(x):

P(x) = (1/5) * x * (x + 4) * (x^2 - 6x + 9)

Next, multiply the first two factors:

x * (x + 4) = x^2 + 4x

Substitute this back into the formula:

P(x) = (1/5) * (x^2 + 4x) * (x^2 - 6x + 9)

Now, multiply the two quadratic factors:

P(x) = (1/5) * (x^4 - 6x^3 + 9x^2 + 4x^3 - 24x^2 + 36x)

Combine like terms:

P(x) = (1/5) * (x^4 - 2x^3 - 15x^2 + 36x)

Finally, distribute the (1/5):

P(x) = (1/5)x^4 - (2/5)x^3 - 3x^2 + (36/5)x

This is the polynomial P(x) in standard form.

To verify our solution, we can check if the polynomial satisfies the given conditions:

  • Roots: We can see that x = 0 and x = -4 are roots because they make the factors x and (x + 4) equal to zero. x = 3 is a root of multiplicity 2 because (x - 3) is squared.

  • Point (5, 36): Let's substitute x = 5 into the expanded form of P(x):

    P(5) = (1/5)(5)^4 - (2/5)(5)^3 - 3(5)^2 + (36/5)(5)

    P(5) = (1/5)(625) - (2/5)(125) - 3(25) + 36

    P(5) = 125 - 50 - 75 + 36

    P(5) = 36

    This confirms that the polynomial passes through the point (5, 36).

In this article, we successfully found a formula for the polynomial P(x) of degree 4, given its roots and a point it passes through. We demonstrated how to use the roots and their multiplicities to construct the factored form of the polynomial and how to determine the leading coefficient using the given point. We also expanded the polynomial to its standard form and verified our solution. This problem highlights the connection between the roots, factors, and coefficients of a polynomial, reinforcing the fundamental concepts of algebra. Understanding these concepts allows us to solve a wide variety of polynomial-related problems and gain a deeper insight into the behavior of polynomial functions.

Polynomial of degree 4, roots of a polynomial, multiplicity of roots, factored form of a polynomial, leading coefficient, polynomial formula, algebraic problem solving, polynomial equation, zeros of a polynomial, verifying solutions.

Q1: How do you find the formula for a polynomial given its roots and a point?

To find the formula for a polynomial P(x) given its roots and a point, first, use the roots and their multiplicities to construct the factored form of the polynomial. This will have the form P(x) = a(x - r1)^m1(x - r2)^m2..., where r1, r2, ... are the roots and m1, m2, ... are their respective multiplicities. Then, use the given point (x, y) that the polynomial passes through to solve for the leading coefficient a by substituting the values into the equation y = P(x). Once you find a, substitute it back into the factored form to obtain the complete formula for P(x).

Q2: What does multiplicity of a root mean?

The multiplicity of a root refers to the number of times a particular root appears as a solution of the polynomial equation. For example, if a root r has a multiplicity of 2, it means the factor (x - r) appears twice in the factored form of the polynomial, like (x - r)^2. A root with multiplicity 1 is called a simple root, while roots with multiplicity greater than 1 are called multiple roots.

Q3: How do you determine the leading coefficient of a polynomial?

To determine the leading coefficient of a polynomial, you need additional information, such as a point (x, y) that the polynomial passes through. Once you have the factored form of the polynomial, P(x) = a(x - r1)^m1(x - r2)^m2..., substitute the x and y values of the given point into the equation. This will give you an equation in terms of a, which you can then solve to find the value of the leading coefficient.

Q4: What is the factored form of a polynomial?

The factored form of a polynomial is an expression of the polynomial as a product of linear factors, each corresponding to a root of the polynomial. The general form is P(x) = a(x - r1)^m1(x - r2)^m2..., where a is the leading coefficient, r1, r2, ... are the roots, and m1, m2, ... are their respective multiplicities. This form is useful for easily identifying the roots of the polynomial and understanding its behavior.

Q5: How do you verify the solution for a polynomial problem?

To verify the solution for a polynomial problem, you can check if the polynomial satisfies all the given conditions. First, verify that the roots match the given roots and their multiplicities. Then, substitute the x value of the given point into the polynomial equation and check if the result matches the given y value. If all conditions are satisfied, the solution is correct.

#Find a Formula for P(x) - A Polynomial Problem

###Understanding Polynomials and their Roots

The problem at hand involves finding the formula for a polynomial P(x) of degree 4, given information about its roots and a point it passes through. This requires a solid understanding of several key concepts in algebra. Let's delve into these foundational ideas before tackling the problem head-on. The foundation of finding the polynomial formula for P(x), given roots and points, lies in understanding the interplay between roots, multiplicity, and factors, these elements shape polynomial behavior and form its core. Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. The degree of a polynomial is determined by the highest power of the variable. Roots, also known as zeros, are the values of x for which the polynomial equals zero. The multiplicity of a root indicates the number of times a particular root appears as a solution, influencing the graph's behavior at that point. For instance, a root with multiplicity 2 touches the x-axis but does not cross it, while a root with multiplicity 1 crosses the x-axis.

The relationship between roots and factors is fundamental. If r is a root of a polynomial P(x), then (x - r) is a factor of P(x). This means that P(x) can be written as (x - r) multiplied by another polynomial. The converse is also true: if (x - r) is a factor of P(x), then r is a root of P(x). This connection allows us to construct the factored form of a polynomial when we know its roots. The concept of factored form simplifies polynomial representation. It expresses a polynomial as a product of linear factors, each corresponding to a root. The general form is P(x) = a(x - r1)(x - r2)...(x - rn), where a is the leading coefficient and r1, r2, ..., rn are the roots. This form directly reveals the roots and their multiplicities, providing a clear picture of the polynomial's zeros and their impact on its graph.

###Applying the Concepts to the Problem

Now, let's apply these concepts to the specific problem. We are given that P(x) is a polynomial of degree 4, which means it has at most 4 roots (counting multiplicities). We know that P(x) has a root of multiplicity 2 at x = 3. This means that the factor (x - 3) appears twice in the factored form of P(x), so we have (x - 3)^2. We also know that P(x) has roots of multiplicity 1 at x = 0 and x = -4. This contributes the factors x and (x + 4), respectively. Combining these factors, we can write P(x) in the following form:

P(x) = a * x * (x + 4) * (x - 3)^2

Here, a is the leading coefficient, which we need to determine. The leading coefficient plays a crucial role in determining the polynomial's overall shape and vertical stretch or compression. It is the coefficient of the term with the highest power of x. In our case, the leading coefficient a is what we need to find to fully define P(x). The fact that P(x) passes through the point (5, 36) gives us the information needed to find a. This point provides a specific x and y value that satisfies the equation P(x) = y. By substituting these values into the equation, we can solve for a.

###Solving for the Leading Coefficient and the Polynomial Formula

We know that P(5) = 36. Let's substitute x = 5 into the equation and solve for a:

36 = a * 5 * (5 + 4) * (5 - 3)^2

Simplifying this equation, we get:

36 = a * 5 * 9 * (2)^2

36 = a * 5 * 9 * 4

36 = 180a

Dividing both sides by 180, we find:

a = 36 / 180 = 1 / 5

So, the leading coefficient a is 1/5. Now that we have the leading coefficient, we can write the complete formula for P(x):

P(x) = (1/5) * x * (x + 4) * (x - 3)^2

This is the formula for the polynomial of degree 4 that satisfies all the given conditions. This polynomial formula is the culmination of our understanding of roots, multiplicities, factors, and leading coefficients. It provides a complete description of P(x) and allows us to predict its behavior for any given value of x.

###Expanding and Verifying the Polynomial (Optional)

While the factored form is often the most useful for analyzing the roots, we can also expand the polynomial to obtain its standard form. This involves multiplying out the factors and combining like terms. First, let's expand the squared term:

(x - 3)^2 = x^2 - 6x + 9

Now, substitute this back into the formula for P(x):

P(x) = (1/5) * x * (x + 4) * (x^2 - 6x + 9)

Next, multiply the first two factors:

x * (x + 4) = x^2 + 4x

Substitute this back into the formula:

P(x) = (1/5) * (x^2 + 4x) * (x^2 - 6x + 9)

Now, multiply the two quadratic factors:

P(x) = (1/5) * (x^4 - 6x^3 + 9x^2 + 4x^3 - 24x^2 + 36x)

Combine like terms:

P(x) = (1/5) * (x^4 - 2x^3 - 15x^2 + 36x)

Finally, distribute the (1/5):

P(x) = (1/5)x^4 - (2/5)x^3 - 3x^2 + (36/5)x

This is the polynomial P(x) in standard form. To verify the solution, we can check if it satisfies the given conditions. The factored form directly shows the roots at x = 0, x = -4, and x = 3 (with multiplicity 2). We can also check if the polynomial passes through the point (5, 36) by substituting x = 5 into the expanded form:

P(5) = (1/5)(5)^4 - (2/5)(5)^3 - 3(5)^2 + (36/5)(5)

P(5) = 125 - 50 - 75 + 36

P(5) = 36

This confirms that our solution is correct.

###Conclusion: The Power of Polynomial Factorization and Root Analysis

In conclusion, we successfully found the formula for P(x), a polynomial of degree 4, by leveraging information about its roots, their multiplicities, and a point it passes through. We demonstrated how the factored form of a polynomial is directly related to its roots and how the leading coefficient can be determined using a given point. This problem illustrates the powerful connection between algebra and problem-solving. The process of finding P(x) showcases the elegance of mathematical reasoning and the interconnectedness of core algebraic concepts. Understanding these principles equips us to tackle a wide range of polynomial-related challenges. By understanding these relationships, we can construct and analyze polynomials effectively, opening doors to solving a variety of mathematical problems.