Welcome to our exploration of quadratic polynomial zeroes! Understanding these zeroes is essential when working with quadratic polynomials, as they represent the values of ( x ) for which the polynomial equals zero. In this post, we will delve into a specific quadratic polynomial problem where we determine its zeroes based on given points. We will analyze how these zeroes influence the graph of the polynomial and clarify how to solve for them, enriching our grasp of polynomial roots.

As we navigate through this topic, we will utilize the quadratic polynomial ( p(x) ) that passes through three distinct points. Specifically, two of these points will provide us with immediate insights into the zeroes. The great thing about examining quadratic polynomial zeroes is that it not only reinforces foundational concepts in algebra but also enhances our problem-solving skills. By the end of this discussion, you will see how determining the zeroes of a quadratic polynomial can significantly impact your understanding of its behavior and graphical representation. So, let’s get started and uncover the fascinating world of quadratic polynomial zeroes together!

“Success is not final, failure is not fatal: it is the courage to continue that counts” – Winston Churchill

In this blog post, we will explore a quadratic polynomial problem involving zeroes. The quadratic polynomial ## p(x) ## passes through three points, and we are tasked with determining its zeroes. This exercise not only reinforces our understanding of quadratic functions but also aids in mastering polynomial roots.

Problem Statement

The given points where the quadratic polynomial passes through are ##(1, 0)##, ##(7, 0)##, and ##(3, -8)##. To find the zeroes of the polynomial, we note that the zeroes are the values of ## x ## where ## p(x) = 0 ##.

From the provided points, ##(1, 0)## and ##(7, 0)## indicate that ## x = 1 ## and ## x = 7 ## are zeroes of the polynomial, as both points yield ## p(x) = 0 ##. The challenge lies in constructing the polynomial and confirming these zeroes analytically.

Solution

Understanding Quadratic Polynomials

A quadratic polynomial can generally be expressed in the standard form as follows:

### p(x) = ax^2 + bx + c ###

where ## a, b, ## and ## c ## are constants and \( a eq 0 \). The properties of quadratic polynomials dictate that they can have at most two zeroes and can be determined using the roots derived from the equation set to zero, ## ax^2 + bx + c = 0 ##. The zeroes of the polynomial are crucial in understanding its graph, as they represent points where the graph intersects the x-axis.

Formulating the Quadratic Polynomial

Since we know two zeroes, ## x = 1 ## and ## x = 7 ##, we can express the quadratic polynomial in its factored form:

### p(x) = a(x – 1)(x – 7) ###

Here, ## a ## is a leading coefficient that will dictate the shape of the parabola. To find ## a ##, we can substitute the third point, ## (3, -8) ##, to obtain a unique polynomial that satisfies all provided points.

Substituting for Coefficient ‘a’

By substituting ## x = 3 ## and ## p(3) = -8 ## into our polynomial form, we have:

###-8 = a(3 – 1)(3 – 7)###

Simplifying gives:

###-8 = a(2)(-4)###

Thus, we arrive at:

### -8 = -8a ###

which provides the value for ## a ##:

###a = 1###.

Hence, we obtain the quadratic polynomial as:

###p(x) = (x – 1)(x – 7)###

Calculating the Quadratic Polynomial

Expanding ## p(x) ##, we have:

###p(x) = x^2 – 8x + 7###

Our polynomial is now in standard form. The roots we calculated originally lead us to assert that the zeroes of this polynomial are located at ## x = 1 ## and ## x = 7 ## due to their representation within the factored form. The significance of this lies not just in confirming the zeroes but also in understanding how these roots form the graph of the polynomial function.

Final Solution: Verifying the Zeroes

The confirmed zeroes of the polynomial ## p(x) ## are indeed 1 and 7, which correspond to options C) 1, 7. These points indicate where the quadratic function intersects the x-axis, aligning perfectly with the original points provided in the problem statement, reflecting the properties of quadratic equations efficiently.

Thus, the confirmation of the result shows that the polynomial aligns with our understanding of quadratic polynomials and their behaviors through calculated roots and corresponding coordinates.

Conclusion

To summarize, the quadratic polynomial ## p(x) = (x – 1)(x – 7) ## successfully passes through the specified points and confirms that the zeroes are at ## x = 1 ## and ## x = 7 ##. This illustrates not only the method for finding zeroes but also enhances our comprehension of how quadratic functions behave. Quadratic polynomial zeroes play a vital role in determining the attributes and characteristics of polynomial graphs.

As we observe, solving for zeroes and understanding quadratic functions can be greatly enhanced through systematic approaches and verification of properties, ensuring accurate and educational experiences while exploring mathematics.

Some more Additional Numerical Problems similar to the above ones are illustrated below

1. ###p(x) = a(x – 2)(x – 4)###

Given points (2, 0) and (4, 0), find the expression of the quadratic polynomial passing through point (3, -3).

Substituting (3, -3) in our polynomial form gives: -3 = a(3 – 2)(3 – 4), leading to a = 3.

Thus, the polynomial is: ##p(x) = 3(x – 2)(x – 4)##.

2. ###p(x) = b(x – 0)(x + 6)###

Find the polynomial that passes through (0, 0) and (6, 0) also passing through (2, -12).

By substituting (2, -12), we arrive at: -12 = b(2)(-4), leading to b = 1.5.

The resulting polynomial is: ##p(x) = 1.5(x)(x + 6)##.

3. ###P(x) = c(x – 1)(x – 3)###

For points (1, 0) and (3, 0), and also (2, -1), find c.

Substituting gives: -1 = c(1)(-1), thus c = 1.

The polynomial is: ##P(x) = (x – 1)(x – 3)##.

4. ###p(x) = d(x – 5)(x – 8)###

Given (5, 0) and (8, 0), find d using the point (7, -4).

Substituting gives: -4 = d(2)(-1), hence d = 2.

The polynomial becomes: ##p(x) = 2(x – 5)(x – 8)##.

5. ###p(x) = e(x – 9)(x + 1)###

Using zeroes (9, 0) and (-1, 0) while satisfying (0, -9) to find e.

Substituting gives: -9 = e(-9)(1), leading to e = 1.

Thus, the polynomial is: ##p(x) = (x – 9)(x + 1)##.

6. ###p(x) = f(x – 4)(x + 2)###

For zeroes (4, 0) and (-2, 0), find f using the point (1, -15).

Solving gives: -15 = f(-3)(3), which leads to f = 5.

The resulting polynomial is: ##p(x) = 5(x – 4)(x + 2)##.

7. ###p(x) = g(x + 3)(x – 3)###

With zeroes at (-3, 0) and (3, 0), pass through (0, -9) to find g.

So, -9 = g(-3)(3), where g comes to 1.

Thus, the polynomial yields: ##p(x) = (x + 3)(x – 3)##.

8. ###p(x) = h(x – 2)(x – 5)###

From (2, 0), (5, 0), and (3, -3) find h.

-3 = h(1)(-2), thus h = 1.5.

The polynomial becomes: ##p(x) = 1.5(x – 2)(x – 5)##.

9. ###p(x) = j(x – 10)(x – 12)###

Find the quadratic polynomial passing through (10, 0), (12, 0), and (11, -4).

It gives: -4 = j(1)(-1), hence j = 4.

The polynomial is: ##p(x) = 4(x – 10)(x – 12)##.

10. ###p(x) = k(x – 6)(x – 7)###

For zeroes at (6, 0) and (7, 0), passing also through (6.5, -2). Find k.

Thus, -2 = k(-0.5)(-0.5), which leads to k = 8.

Final polynomial: ##p(x) = 8(x – 6)(x – 7)##.

These similar problems showcase how to find quadratic polynomials given specific points and zeroes. Each polynomial is formulated to ensure a thorough understanding of how to manipulate quadratic functions, essential in both mathematical and real-world applications.

In our journey through understanding quadratic polynomial zeroes, we have highlighted their pivotal role in determining the behavior of quadratic functions. These zeroes not only signify the points of intersection with the x-axis but are also integral in analyzing the graph of the polynomial. With the fundamental takeaways from our problem-solving session, remember that identifying zeroes helps us refine our knowledge of polynomial roots.

Quadratic polynomial zeroes are more than mere solutions; they provide deep insights into the characteristics of parabolas—such as where they open, the vertex’s location, and the dress of the curve. Moreover, practicing with various polynomial forms strengthens your ability to tackle more complex mathematical challenges involving higher-degree polynomials.

• Always validate your polynomial roots to ensure accuracy.
• Explore different forms of quadratic equations to gain proficiency in deriving zeroes.
• Understand the role of the leading coefficient in impacting the graph’s orientation.
• Utilize external resources or graphing tools to visualize quadratic functions for a better understanding of zeroes.
• Utilize external resources or graphing tools to visualize quadratic functions for a better understanding of zeroes.

Thus, mastering quadratic polynomial zeroes not only serves your academic pursuits but also enhances your mathematical reasoning in real-world problem-solving scenarios. Keep practicing, and embrace the fascinating world of quadratic polynomials!