The Graph Of A Quadratic Polynomial $p(x)$ Passes Through The Points $(-6,0)$, \$(0,-30)$[/tex\], $(4,-20)$, And $(6,0)$. The Zeroes Of The Polynomial Are:
Introduction
Quadratic Polynomials and Their Graphs A quadratic polynomial is a polynomial of degree two, which means the highest power of the variable is two. The general form of a quadratic polynomial is $p(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be equal to zero. The graph of a quadratic polynomial is a parabola, which is a U-shaped curve. In this article, we will discuss how to find the zeroes of a quadratic polynomial given its graph passes through certain points.
The Problem
We are given that the graph of a quadratic polynomial $p(x)$ passes through the points $(-6,0)$, $(0,-30)$, $(4,-20)$, and $(6,0)$. We need to find the zeroes of the polynomial, which are the values of $x$ where the polynomial equals zero.
Understanding the Graph
The graph of a quadratic polynomial is a parabola, which is a U-shaped curve. The graph passes through the points $(-6,0)$, $(0,-30)$, $(4,-20)$, and $(6,0)$. This means that the graph touches the x-axis at the points $(-6,0)$ and $(6,0)$, and it also passes through the point $(0,-30)$ on the y-axis.
Finding the Zeroes
To find the zeroes of the polynomial, we need to find the values of $x$ where the polynomial equals zero. Since the graph passes through the points $(-6,0)$ and $(6,0)$, we know that the polynomial has zeroes at $x = -6$ and $x = 6$.
Using the Factor Theorem
The factor theorem states that if a polynomial $p(x)$ has a zero at $x = r$, then $(x - r)$ is a factor of the polynomial. Since we know that the polynomial has zeroes at $x = -6$ and $x = 6$, we can write the polynomial as $p(x) = a(x + 6)(x - 6)$, where $a$ is a constant.
Finding the Value of $a$
We are given that the graph passes through the point $(0,-30)$. We can substitute this point into the polynomial to find the value of $a$. Plugging in $x = 0$ and $p(x) = -30$, we get:
Simplifying, we get:
Dividing both sides by $-36$, we get:
Writing the Polynomial
Now that we have found the value of $a$, we can write the polynomial as:
Conclusion
In this article, we discussed how to find the zeroes of a quadratic polynomial given its graph passes through certain points. We used the factor theorem to write the polynomial as a product of linear factors, and we found the value of $a$ by substituting a point on the graph into the polynomial. We concluded that the zeroes of the polynomial are $x = -6$ and $x = 6$.
The Final Answer
The final answer is:
Introduction
In our previous article, we discussed how to find the zeroes of a quadratic polynomial given its graph passes through certain points. In this article, we will provide a Q&A guide to help you understand the concept of quadratic polynomial zeroes better.
Q: What is a quadratic polynomial?
A: A quadratic polynomial is a polynomial of degree two, which means the highest power of the variable is two. The general form of a quadratic polynomial is $p(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be equal to zero.
Q: What is the graph of a quadratic polynomial?
A: The graph of a quadratic polynomial is a parabola, which is a U-shaped curve. The graph of a quadratic polynomial can be either upward-facing or downward-facing, depending on the value of $a$.
Q: How do I find the zeroes of a quadratic polynomial?
A: To find the zeroes of a quadratic polynomial, you need to find the values of $x$ where the polynomial equals zero. You can use the factor theorem to write the polynomial as a product of linear factors, and then find the values of $x$ that make each factor equal to zero.
Q: What is the factor theorem?
A: The factor theorem states that if a polynomial $p(x)$ has a zero at $x = r$, then $(x - r)$ is a factor of the polynomial. This means that if you know that a polynomial has a zero at a certain value of $x$, you can write the polynomial as a product of linear factors, with one of the factors being $(x - r)$.
Q: How do I use the factor theorem to find the zeroes of a quadratic polynomial?
A: To use the factor theorem to find the zeroes of a quadratic polynomial, you need to write the polynomial as a product of linear factors, and then find the values of $x$ that make each factor equal to zero. For example, if you know that a quadratic polynomial has zeroes at $x = -6$ and $x = 6$, you can write the polynomial as $p(x) = a(x + 6)(x - 6)$, where $a$ is a constant.
Q: How do I find the value of $a$ in a quadratic polynomial?
A: To find the value of $a$ in a quadratic polynomial, you need to substitute a point on the graph into the polynomial and solve for $a$. For example, if you know that a quadratic polynomial passes through the point $(0,-30)$, you can substitute $x = 0$ and $p(x) = -30$ into the polynomial and solve for $a$.
Q: What are the zeroes of a quadratic polynomial?
A: The zeroes of a quadratic polynomial are the values of $x$ where the polynomial equals zero. These values are also known as the roots of the polynomial.
Q: Can a quadratic polynomial have more than two zeroes?
A: No, a quadratic polynomial can have at most two zeroes. This is because a quadratic polynomial is of degree two, which means it has at most two roots.
Q: Can a quadratic polynomial have no zeroes?
A: Yes, a quadratic polynomial can have no zeroes. This occurs when the polynomial has no real roots, which means it does not intersect the x-axis.
Conclusion
In this article, we provided a Q&A guide to help you understand the concept of quadratic polynomial zeroes better. We discussed how to find the zeroes of a quadratic polynomial using the factor theorem, and how to find the value of $a$ in a quadratic polynomial. We also answered common questions about quadratic polynomial zeroes, including how to find the zeroes of a quadratic polynomial and how to determine if a quadratic polynomial has any zeroes.
The Final Answer
The final answer is: