Solving Problems On Quadratic Polynomials And Their Zeroes

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In the realm of mathematics, quadratic polynomials hold a significant place. These polynomials, characterized by their highest degree being two, frequently appear in various mathematical and real-world applications. Understanding their properties, especially the concept of zeroes, is crucial for solving related problems. This article delves into specific problems concerning quadratic polynomials and their zeroes, offering detailed solutions and explanations to enhance comprehension. We will explore how to determine the value of unknown coefficients within these polynomials when given information about their zeroes. Let's embark on this mathematical journey to unravel the intricacies of quadratic polynomials.

Understanding Quadratic Polynomials

A quadratic polynomial is a polynomial of degree two. The general form of a quadratic polynomial is expressed as:

f(x) = ax^2 + bx + c

where a, b, and c are constants, and a ≠ 0. The zeroes of a quadratic polynomial are the values of x for which f(x) = 0. These zeroes are also known as the roots of the quadratic equation ax^2 + bx + c = 0. A quadratic polynomial can have at most two real zeroes, which can be found using various methods, including factoring, completing the square, or the quadratic formula.

The quadratic formula is a fundamental tool for finding the zeroes of any quadratic polynomial. It is given by:

x = [-b ± √(b^2 - 4ac)] / 2a

where a, b, and c are the coefficients of the quadratic polynomial ax^2 + bx + c. The expression b^2 - 4ac is known as the discriminant, which provides valuable information about the nature of the roots. If the discriminant is positive, the polynomial has two distinct real roots. If it is zero, the polynomial has one real root (a repeated root). If it is negative, the polynomial has two complex roots.

The relationship between the zeroes and the coefficients of a quadratic polynomial is another key concept. If α and β are the zeroes of the quadratic polynomial ax^2 + bx + c, then:

  • Sum of zeroes: α + β = -b/a
  • Product of zeroes: αβ = c/a

These relationships are incredibly useful for verifying solutions and for constructing quadratic polynomials when the zeroes are known. For instance, if you know the sum and product of the zeroes, you can directly form the quadratic polynomial using the formula:

f(x) = k[x^2 - (sum of zeroes)x + (product of zeroes)]

where k is any non-zero constant. This formula is derived from the factored form of the quadratic polynomial, which can be written as a(x - α)(x - β), where α and β are the zeroes.

Understanding these fundamental concepts is essential for tackling problems involving quadratic polynomials. The ability to identify the coefficients, apply the quadratic formula, and use the relationships between zeroes and coefficients are critical skills in algebra and beyond. In the subsequent sections, we will apply these concepts to solve specific problems, demonstrating their practical application and reinforcing the theoretical knowledge.

Problem 1: Finding k when one zero of egin{document}$ (k-1)x^2 + kx + 1 $\end{document} is -3

Let's delve into the first problem: **If one zero of the quadratic polynomial egindocument}$ (k-1)x^2 + kx + 1 $\end{document} is -3, then the value of k is** This problem requires us to find the value of k given that one of the zeroes of the quadratic polynomial egin{document$ (k-1)x^2 + kx + 1 $\end{document} is -3. The core concept here is that if -3 is a zero of the polynomial, then substituting x = -3 into the polynomial will make the expression equal to zero. This is because, by definition, a zero of a polynomial is a value of x that makes the polynomial evaluate to zero.

To solve this, we substitute x = -3 into the quadratic polynomial:

egin{document}(k−1)(−3)2+k(−3)+1=0(k-1)(-3)^2 + k(-3) + 1 = 0\

This substitution transforms the polynomial equation into a linear equation in terms of k. Now, we simplify and solve for k:

egin{document}(k−1)(9)−3k+1=0(k-1)(9) - 3k + 1 = 0\

Expanding the terms, we get:

egin{document}9k−9−3k+1=09k - 9 - 3k + 1 = 0\

Combining like terms, we have:

egin{document}6k−8=06k - 8 = 0\

Adding 8 to both sides of the equation:

egin{document}6k=86k = 8\

Finally, dividing both sides by 6:

egin{document}k=8/6=4/3k = 8/6 = 4/3\

Therefore, the value of k is 4/3. This corresponds to option (a) in the given choices. This problem illustrates a fundamental principle in dealing with polynomials: knowing a zero of a polynomial allows you to find relationships between its coefficients. By substituting the zero into the polynomial, you can create an equation that involves the unknown coefficients, which can then be solved using algebraic techniques.

This type of problem is commonly encountered in algebra and is a basic yet crucial skill for understanding polynomial behavior. It demonstrates how the zeroes of a polynomial are intrinsically linked to its coefficients, providing a powerful tool for solving various mathematical problems. The ability to manipulate polynomial expressions and solve for unknown variables is a core competency in mathematics, and this problem serves as a practical example of how to apply these skills.

Problem 2: Finding k when -1 is a zero of egin{document}$ kx^2 - 4x + k $\end{document}

Let's tackle the second problem: **If -1 is a zero of the polynomial egindocument}$ kx^2 - 4x + k $\end{document}, the value of k is** This problem is similar to the first one, but it reinforces the concept of using zeroes to find unknown coefficients. Here, we are given that -1 is a zero of the quadratic polynomial egin{document$ kx^2 - 4x + k $\end{document}, and our goal is to determine the value of k. The key principle remains the same: if -1 is a zero, then substituting x = -1 into the polynomial will result in the polynomial evaluating to zero.

We substitute x = -1 into the quadratic polynomial:

egin{document}k(−1)2−4(−1)+k=0k(-1)^2 - 4(-1) + k = 0\

This substitution yields an equation in terms of k. Now, we simplify and solve for k:

egin{document}k(1)+4+k=0k(1) + 4 + k = 0\

Combining like terms, we get:

egin{document}2k+4=02k + 4 = 0\

Subtracting 4 from both sides of the equation:

egin{document}2k=−42k = -4\

Finally, dividing both sides by 2:

egin{document}k=−2k = -2\

Thus, the value of k is -2. This corresponds to option (b) in the given choices. This problem further solidifies the understanding of the relationship between the zeroes and coefficients of a polynomial. By using the fact that a zero of a polynomial makes the polynomial equal to zero, we can create and solve equations to find unknown parameters.

This type of problem is a common exercise in algebra and is crucial for developing proficiency in polynomial manipulation. It highlights the importance of understanding the definition of a zero and how to apply it in practical problem-solving scenarios. The ability to substitute values into polynomial expressions and solve the resulting equations is a fundamental skill that is widely used in various mathematical contexts.

Furthermore, this problem underscores the symmetry that can exist within quadratic polynomials. In this case, the coefficient of the egin{document}$ x^2 $\end{document} term and the constant term are both k. This symmetry can sometimes provide additional insights or simplifications when solving related problems. Recognizing such patterns can be a valuable asset in mathematical problem-solving.

Conclusion

In conclusion, the problems discussed in this article provide a clear understanding of how to find unknown coefficients in quadratic polynomials when given information about their zeroes. By substituting the zeroes into the polynomial equation and solving for the unknowns, we can effectively tackle these types of problems. The core concept is that a zero of a polynomial makes the polynomial equal to zero, which allows us to create an equation involving the unknown coefficients.

These problems are not only valuable for academic purposes but also lay the foundation for more advanced mathematical concepts. The ability to manipulate polynomials, solve equations, and understand the relationship between zeroes and coefficients are essential skills in various fields, including engineering, physics, and computer science. Mastering these concepts will undoubtedly enhance one's problem-solving abilities and mathematical acumen.

The first problem demonstrated the direct application of substituting a zero into the polynomial and solving for the unknown coefficient k. The second problem reinforced this concept, showcasing its versatility in handling different polynomial expressions. Both problems highlighted the importance of careful algebraic manipulation and the ability to simplify equations to arrive at the correct solution.

Moreover, these examples emphasize the significance of understanding the fundamental definitions and principles in mathematics. The definition of a zero of a polynomial is the cornerstone of the solution process, and without a clear understanding of this concept, solving such problems would be significantly more challenging. Therefore, a solid foundation in basic mathematical principles is crucial for success in more advanced topics.

In summary, this exploration of quadratic polynomials and their zeroes has provided valuable insights into problem-solving techniques and the underlying mathematical concepts. By understanding the relationship between zeroes and coefficients, we can effectively solve a wide range of problems involving polynomials. The skills acquired through these exercises will undoubtedly prove beneficial in future mathematical endeavors and beyond.