Polynomial Function Operations A Step-by-Step Guide

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In the realm of mathematics, polynomial functions stand as fundamental building blocks. They are expressions comprising variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. This article delves into the intricacies of polynomial functions, focusing on performing various operations such as addition and subtraction. We will explore these concepts through a detailed example, providing a step-by-step guide to mastering these operations.

Defining Polynomial Functions

Before we delve into the operations, let's define the functions we will be working with:

  • h(x) = -14x² + 16x + 5
  • f(x) = 13x² - 6x - 3
  • g(x) = 18x² + 7x - 4

These are quadratic functions, a specific type of polynomial function where the highest power of the variable x is 2. Each function is expressed in the standard form of a quadratic equation: ax² + bx + c, where a, b, and c are constants. Understanding this form is crucial for performing operations on these functions.

Addition of Polynomial Functions

(f + g)(x): Combining f(x) and g(x)

To find (f + g)(x), we add the expressions for f(x) and g(x) together. This involves combining like terms, which are terms with the same variable and exponent. This operation, combining f(x) and g(x), is a fundamental concept in polynomial arithmetic and forms the basis for more complex calculations.

Here’s the step-by-step process:

  1. Write out the expressions for f(x) and g(x):

    f(x) = 13x² - 6x - 3

    g(x) = 18x² + 7x - 4

  2. Add the two expressions:

    (f + g)(x) = (13x² - 6x - 3) + (18x² + 7x - 4)

  3. Combine like terms:

    (f + g)(x) = (13x² + 18x²) + (-6x + 7x) + (-3 - 4)

  4. Simplify:

    (f + g)(x) = 31x² + x - 7

Therefore, (f + g)(x) = 31x² + x - 7. This resultant quadratic function represents the sum of the two original functions. Understanding how to add these functions is vital in various mathematical applications, including calculus and algebraic manipulations. The combination of like terms is a recurring theme in polynomial operations, making it a core skill to master.

(f + h)(x): Combining f(x) and h(x)

Next, let's find (f + h)(x) by adding the expressions for f(x) and h(x). Similar to the previous example, we will combine like terms to simplify the expression. The function f(x), a crucial component in this operation, is combined with h(x) to produce a new polynomial function. This process is a cornerstone of polynomial manipulation and has wide-ranging applications in mathematics and engineering.

  1. Write out the expressions for f(x) and h(x):

    f(x) = 13x² - 6x - 3

    h(x) = -14x² + 16x + 5

  2. Add the two expressions:

    (f + h)(x) = (13x² - 6x - 3) + (-14x² + 16x + 5)

  3. Combine like terms:

    (f + h)(x) = (13x² - 14x²) + (-6x + 16x) + (-3 + 5)

  4. Simplify:

    (f + h)(x) = -x² + 10x + 2

Thus, (f + h)(x) = -x² + 10x + 2. This new quadratic function represents the sum of f(x) and h(x), illustrating the result of adding two polynomial functions. The concept of like terms is once again central to this operation, reinforcing its importance in polynomial arithmetic. This skill is particularly useful when dealing with complex algebraic expressions and is often applied in fields such as physics and computer science.

Subtraction of Polynomial Functions

Subtraction of polynomial functions involves subtracting one polynomial from another. This process also relies on combining like terms, but it is crucial to distribute the negative sign correctly.

(f - g)(x): Subtracting g(x) from f(x)

To find (f - g)(x), we subtract the expression for g(x) from f(x). This involves distributing the negative sign across all terms in g(x) before combining like terms. The accurate distribution of the negative sign is paramount in polynomial subtraction, and errors in this step can lead to incorrect results. Understanding this process is essential for various mathematical and engineering applications.

  1. Write out the expressions for f(x) and g(x):

    f(x) = 13x² - 6x - 3

    g(x) = 18x² + 7x - 4

  2. Subtract g(x) from f(x):

    (f - g)(x) = (13x² - 6x - 3) - (18x² + 7x - 4)

  3. Distribute the negative sign:

    (f - g)(x) = 13x² - 6x - 3 - 18x² - 7x + 4

  4. Combine like terms:

    (f - g)(x) = (13x² - 18x²) + (-6x - 7x) + (-3 + 4)

  5. Simplify:

    (f - g)(x) = -5x² - 13x + 1

Therefore, (f - g)(x) = -5x² - 13x + 1. The result is a new quadratic function obtained by subtracting g(x) from f(x). This type of subtraction is fundamental in many areas of mathematics, such as finding the difference between two curves or analyzing residual functions in statistical models. The accurate manipulation of negative signs is a critical skill emphasized here.

(g - f)(x): Subtracting f(x) from g(x)

Now, let's find (g - f)(x) by subtracting f(x) from g(x). This operation is the reverse of the previous one and will yield a different result, highlighting the importance of the order of subtraction. The order of subtraction significantly affects the outcome, underscoring the non-commutative nature of this operation in polynomial arithmetic.

  1. Write out the expressions for f(x) and g(x):

    f(x) = 13x² - 6x - 3

    g(x) = 18x² + 7x - 4

  2. Subtract f(x) from g(x):

    (g - f)(x) = (18x² + 7x - 4) - (13x² - 6x - 3)

  3. Distribute the negative sign:

    (g - f)(x) = 18x² + 7x - 4 - 13x² + 6x + 3

  4. Combine like terms:

    (g - f)(x) = (18x² - 13x²) + (7x + 6x) + (-4 + 3)

  5. Simplify:

    (g - f)(x) = 5x² + 13x - 1

Thus, (g - f)(x) = 5x² + 13x - 1. This result is the negation of (f - g)(x), illustrating the effect of reversing the order of subtraction. Understanding this concept is vital in various mathematical contexts, including linear algebra and differential equations. The contrast between (f - g)(x) and (g - f)(x) clearly demonstrates the importance of the order of operations.

(g - h)(x): Subtracting h(x) from g(x)

Next, we find (g - h)(x) by subtracting h(x) from g(x). This operation is similar to the previous subtraction examples, emphasizing the importance of correctly distributing the negative sign and combining like terms. The function h(x), an integral part of this calculation, is subtracted from g(x) to yield a new quadratic function. This skill is particularly useful in advanced mathematical modeling and analysis.

  1. Write out the expressions for g(x) and h(x):

    g(x) = 18x² + 7x - 4

    h(x) = -14x² + 16x + 5

  2. Subtract h(x) from g(x):

    (g - h)(x) = (18x² + 7x - 4) - (-14x² + 16x + 5)

  3. Distribute the negative sign:

    (g - h)(x) = 18x² + 7x - 4 + 14x² - 16x - 5

  4. Combine like terms:

    (g - h)(x) = (18x² + 14x²) + (7x - 16x) + (-4 - 5)

  5. Simplify:

    (g - h)(x) = 32x² - 9x - 9

Therefore, (g - h)(x) = 32x² - 9x - 9. This resulting quadratic function demonstrates the outcome of subtracting h(x) from g(x). This type of operation is commonly used in optimization problems and curve fitting, where differences between functions need to be precisely determined. Careful attention to detail is essential in these calculations to ensure accuracy.

(h - g)(x): Subtracting g(x) from h(x)

Now, let's calculate (h - g)(x) by subtracting g(x) from h(x). This operation further illustrates the significance of the order of subtraction and provides additional practice in distributing negative signs correctly. The subtraction of g(x) from h(x), a crucial operation in this context, yields a different result compared to subtracting h(x) from g(x). This distinction highlights the importance of precision in mathematical operations.

  1. Write out the expressions for g(x) and h(x):

    h(x) = -14x² + 16x + 5

    g(x) = 18x² + 7x - 4

  2. Subtract g(x) from h(x):

    (h - g)(x) = (-14x² + 16x + 5) - (18x² + 7x - 4)

  3. Distribute the negative sign:

    (h - g)(x) = -14x² + 16x + 5 - 18x² - 7x + 4

  4. Combine like terms:

    (h - g)(x) = (-14x² - 18x²) + (16x - 7x) + (5 + 4)

  5. Simplify:

    (h - g)(x) = -32x² + 9x + 9

Thus, (h - g)(x) = -32x² + 9x + 9. This quadratic function is the negative of (g - h)(x), emphasizing the effect of reversing the order of subtraction. This concept is widely used in calculus, particularly when dealing with derivatives and integrals. The relationship between (h - g)(x) and (g - h)(x) provides a clear illustration of the properties of subtraction.

(h - f)(x): Subtracting f(x) from h(x)

Finally, we find (h - f)(x) by subtracting f(x) from h(x). This operation provides another opportunity to practice polynomial subtraction and reinforces the importance of distributing the negative sign. Subtracting f(x) from h(x), a key step in polynomial manipulation, results in a new quadratic function. This process is fundamental in various mathematical applications, including curve analysis and optimization problems.

  1. Write out the expressions for f(x) and h(x):

    h(x) = -14x² + 16x + 5

    f(x) = 13x² - 6x - 3

  2. Subtract f(x) from h(x):

    (h - f)(x) = (-14x² + 16x + 5) - (13x² - 6x - 3)

  3. Distribute the negative sign:

    (h - f)(x) = -14x² + 16x + 5 - 13x² + 6x + 3

  4. Combine like terms:

    (h - f)(x) = (-14x² - 13x²) + (16x + 6x) + (5 + 3)

  5. Simplify:

    (h - f)(x) = -27x² + 22x + 8

Therefore, (h - f)(x) = -27x² + 22x + 8. This resulting quadratic function completes our exploration of subtraction operations between the given functions. This operation is crucial in fields such as control systems and signal processing, where understanding the differences between functions is essential. Polynomial subtraction, as demonstrated in this example, is a versatile tool in mathematical analysis.

Conclusion

In this article, we have thoroughly explored the addition and subtraction of polynomial functions. By working through the examples with functions h(x), f(x), and g(x), we have demonstrated the step-by-step processes involved in these operations. These skills are fundamental in algebra and are essential for more advanced mathematical studies. The ability to add and subtract polynomial functions is a core competency in mathematics and is widely applied in various scientific and engineering disciplines. Mastering these operations will provide a solid foundation for tackling more complex mathematical problems and real-world applications.

Polynomial Functions, Addition, Subtraction, Quadratic Functions, Like Terms, Distribution, Algebraic Expressions, Mathematics, Equations, Expressions

Find (f + g)(x), (f - g)(x), (g - f)(x), (f + h)(x), (g - h)(x), (h - g)(x), and (h - f)(x) given h(x) = -14x² + 16x + 5, f(x) = 13x² - 6x - 3, g(x) = 18x² + 7x - 4. How to solve polynomial operations.