Polynomial Formula A Cubic Equation With Given Roots And Intercepts

July 10, 2025 by Scholario Team 68 views

Unveiling the Formula for a Cubic Polynomial with Specific Roots and Y-intercept

In the fascinating world of polynomial functions, we often encounter scenarios where we need to determine the equation of a polynomial given specific information about its roots and intercepts. This article delves into a classic problem of this nature, focusing on a cubic polynomial, P(x), characterized by a root of multiplicity 2 at x = 4, a root of multiplicity 1 at x = -5, and a y-intercept at (0, -48). Our mission is to uncover the formula for P(x), and we will embark on this journey by leveraging our understanding of polynomial roots, multiplicities, and the significance of the y-intercept.

Deciphering the Roots and Multiplicities

The foundation of our quest lies in the roots of the polynomial. Recall that a root of a polynomial is a value of x that makes the polynomial equal to zero. In our case, we are given two distinct roots: x = 4 and x = -5. However, the problem statement introduces a crucial detail: the root at x = 4 has a multiplicity of 2, while the root at x = -5 has a multiplicity of 1. This concept of multiplicity plays a pivotal role in shaping the behavior of the polynomial function.

Multiplicity, in essence, tells us how many times a particular root appears as a factor of the polynomial. A root with multiplicity 2, like x = 4 in our case, implies that the factor (x - 4) appears twice in the polynomial's factored form. Similarly, a root with multiplicity 1, like x = -5, indicates that the factor (x + 5) appears only once. This understanding of multiplicity is critical because it directly influences the graph of the polynomial near the root. At a root with multiplicity 2, the graph touches the x-axis but does not cross it, while at a root with multiplicity 1, the graph crosses the x-axis.

Given this information, we can construct a preliminary form of our polynomial P(x). Since x = 4 is a root of multiplicity 2, we include the factor (x - 4)^2. And since x = -5 is a root of multiplicity 1, we include the factor (x + 5). Thus, our polynomial can be expressed as P(x) = a(x - 4)^2(x + 5), where a is a constant coefficient that we still need to determine. This constant factor a is crucial because it scales the polynomial vertically, affecting its overall shape and y-intercept. To find the precise value of a, we will utilize the information about the y-intercept.

Harnessing the Power of the Y-intercept

The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when x = 0. In our problem, we are given that the y-intercept of P(x) is (0, -48). This means that when x = 0, P(x) = -48. This piece of information is the key to unlocking the value of the constant coefficient a in our polynomial expression.

To leverage the y-intercept, we substitute x = 0 into our current expression for P(x): P(0) = a(0 - 4)^2(0 + 5). We know that P(0) must equal -48, so we can set up the equation -48 = a(-4)^2(5). Simplifying this equation, we get -48 = a(16)(5), which further simplifies to -48 = 80a. Solving for a, we find that a = -48/80, which reduces to a = -3/5. Now we have successfully determined the value of the leading coefficient a.

With the value of a in hand, we can refine our expression for P(x). We now know that P(x) = (-3/5)(x - 4)^2(x + 5). This is the factored form of our polynomial, and it explicitly shows the roots and their multiplicities. However, to fully unveil the polynomial, it's often beneficial to expand this expression into its standard form, which is a sum of terms with decreasing powers of x. This expansion will give us a more comprehensive view of the polynomial's structure and behavior.

Unveiling the Formula for P(x)

Now that we have P(x) = (-3/5)(x - 4)^2(x + 5), our final step is to expand this expression to obtain the polynomial in its standard form. This involves multiplying out the factors and combining like terms. Let's embark on this algebraic journey.

First, we expand the squared term (x - 4)^2, which gives us (x - 4)(x - 4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16. Now our polynomial expression looks like P(x) = (-3/5)(x^2 - 8x + 16)(x + 5). Next, we multiply the quadratic expression (x^2 - 8x + 16) by (x + 5). This gives us:

x^2(x + 5) = x^3 + 5x^2
-8x(x + 5) = -8x^2 - 40x
16(x + 5) = 16x + 80

Adding these terms together, we get x^3 + 5x^2 - 8x^2 - 40x + 16x + 80 = x^3 - 3x^2 - 24x + 80. So now we have P(x) = (-3/5)(x^3 - 3x^2 - 24x + 80). Finally, we distribute the factor of (-3/5) across the terms inside the parentheses:

(-3/5)x^3 = (-3/5)x^3
(-3/5)(-3x^2) = (9/5)x^2
(-3/5)(-24x) = (72/5)x
(-3/5)(80) = -48

Combining these terms, we arrive at the standard form of our polynomial: P(x) = (-3/5)x^3 + (9/5)x^2 + (72/5)x - 48. This is the formula for the cubic polynomial that satisfies the given conditions: a root of multiplicity 2 at x = 4, a root of multiplicity 1 at x = -5, and a y-intercept at (0, -48).

In conclusion, by carefully analyzing the roots, their multiplicities, and the y-intercept, we have successfully constructed the formula for the cubic polynomial P(x). This process highlights the powerful connection between the roots of a polynomial and its algebraic representation, showcasing how specific information about a polynomial's behavior can lead us to its unique equation. This methodical approach, combining the concepts of roots, multiplicities, and intercepts, provides a robust framework for tackling similar polynomial problems.

Find the formula for the polynomial P(x) of degree 3 that has a root of multiplicity 2 at x = 4, a root of multiplicity 1 at x = -5, and a y-intercept at (0, -48).