Find X And Y Intercepts Of Polynomial Function

Understanding the x- and y-intercepts of a function is a fundamental concept in mathematics, especially when analyzing polynomial functions like $g(x)=(x+1)\left(x^2-10 x+24\right)$. These intercepts give us crucial points where the graph of the function crosses the x-axis and the y-axis, providing valuable insights into the function's behavior and roots. The y-intercept is the point where the graph intersects the y-axis, and it occurs when $x=0$. On the other hand, the x-intercepts are the points where the graph intersects the x-axis, and these occur when $g(x)=0$. Finding these points helps us sketch the graph, solve equations, and understand the overall structure of the function. In this article, we will delve into how to accurately determine the x- and y-intercepts for the given function $g(x)=(x+1)\left(x^2-10 x+24\right)$, exploring the algebraic steps involved and the meaning behind these specific points on the coordinate plane. We'll break down the process of factoring the polynomial and setting the function equal to zero to find its roots, as well as a straightforward method to calculate the y-intercept. This exploration will not only solve the problem at hand but also reinforce the general techniques applicable to any polynomial function.

Calculating the Y-Intercept

Let's start by finding the y-intercept of the function $g(x)=(x+1)\left(x^2-10 x+24\right)$. The y-intercept is the point where the graph of the function crosses the y-axis. This happens when the input value, $x$, is equal to zero. To find the y-intercept, we simply substitute $x=0$ into the function's equation. This is a very direct and simple process for any function, and for polynomial functions, it often results in the constant term of the expanded polynomial. Let's perform this calculation for our specific function: $g(0) = (0+1)\left(0^2 - 10(0) + 24\right)$. Simplifying this expression, we get $g(0) = (1)\left(0 - 0 + 24\right)$, which further simplifies to $g(0) = 1 \times 24$. Therefore, the y-intercept is 24. This means that the graph of the function $g(x)$ will pass through the point $(0, 24)$ on the y-axis. It's important to remember that a function can have at most one y-intercept, as for each input $x$, there can only be one output $g(x)$. This value, 24, is a key characteristic of our function's graph and provides a reference point for visualization. The ease with which we find the y-intercept highlights its role as a constant in the function's definition, unaffected by the variable terms when $x$ is zero.

Determining the X-Intercepts

The x-intercepts, also known as the roots or zeros of the function, are the points where the graph of the function crosses or touches the x-axis. At these points, the value of the function, $g(x)$, is equal to zero. To find the x-intercepts, we need to set the entire function equal to zero and solve for $x$: $g(x) = (x+1)\left(x^2-10 x+24\right) = 0$. This equation is already in a factored form, which makes finding the roots much simpler. The Zero Product Property states that if a product of factors is equal to zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for $x$:

1. First factor: $x+1 = 0$. Subtracting 1 from both sides, we get $x = -1$.
2. Second factor: $x^2 - 10x + 24 = 0$. This is a quadratic equation. We can solve this by factoring, using the quadratic formula, or completing the square. Let's try factoring. We are looking for two numbers that multiply to 24 and add up to -10. These numbers are -4 and -6. So, we can factor the quadratic as $(x-4)(x-6) = 0$.
   • Setting the first part of the factored quadratic to zero: $x-4 = 0$. Adding 4 to both sides gives us $x = 4$.
   • Setting the second part of the factored quadratic to zero: $x-6 = 0$. Adding 6 to both sides gives us $x = 6$.

Therefore, the x-intercepts are $x = -1$, $x = 4$, and $x = 6$. These are the points where the graph of $g(x)$ crosses the x-axis. A polynomial function of degree $n$ can have at most $n$ real roots, and in this case, our cubic polynomial has three distinct real roots, which is the maximum possible. These roots are critical for understanding the intervals where the function is positive or negative, and they play a key role in sketching the graph accurately.

Summarizing the Intercepts

To summarize our findings for the function $g(x)=(x+1)\left(x^2-10 x+24\right)$, we have successfully identified both the x-intercepts and the y-intercept. The y-intercept is found by evaluating the function at $x=0$, which yielded the point $(0, 24)$. This indicates where the graph crosses the vertical axis. The x-intercepts are found by setting the function equal to zero and solving for $x$. By factoring the polynomial, we determined that the function equals zero when $x=-1$, $x=4$, or $x=6$. These are the points where the graph crosses the horizontal axis: $(-1, 0)$, $(4, 0)$, and $(6, 0)$. Combining these points with our understanding of the function's behavior (e.g., its end behavior and multiplicity of roots), we can begin to visualize the shape of its graph. The three distinct x-intercepts tell us that the graph will cross the x-axis at three different locations. The y-intercept gives us a starting point for sketching. This comprehensive understanding of intercepts is a cornerstone of function analysis in algebra.

Conclusion and Further Exploration

In conclusion, for the function $g(x)=(x+1)\left(x^2-10 x+24\right)$, the x-intercepts are $-1, 4,$ and $6$, and the y-intercept is $24$. These points are crucial for understanding the graphical representation and behavior of the function. They identify where the function's graph intersects the coordinate axes, providing key reference points for analysis and sketching. Mastering the techniques for finding intercepts is essential for a solid foundation in algebra and calculus. For further exploration into polynomial functions and their properties, you can refer to valuable resources like Khan Academy's Algebra section, which offers comprehensive lessons and practice problems on topics including factoring, roots, and graphing functions. Additionally, exploring Paul's Online Math Notes can provide deeper insights into polynomial behavior and analysis.