Simplifying (x-7)(x-5) Using FOIL Method And Expressing In Standard Form

In mathematics, simplifying expressions is a fundamental skill. One common task is expanding the product of two binomials. The FOIL method provides a structured approach to this, ensuring that each term in the first binomial is multiplied by each term in the second binomial. This article will guide you through simplifying the expression $(x-7)(x-5)$ using the FOIL method and expressing the result in standard form. Understanding this method is crucial for various algebraic manipulations, including solving equations, factoring polynomials, and simplifying complex expressions. The FOIL method, which stands for First, Outer, Inner, Last, is a mnemonic device for remembering the order in which to multiply the terms in two binomials. By systematically applying this method, we can avoid overlooking any terms and ensure an accurate simplification. Moreover, expressing the simplified expression in standard form provides a clear and concise representation, making it easier to identify the coefficients and degree of the polynomial. This standard form is essential for further algebraic operations and analyses.

Breaking Down the FOIL Method

The FOIL method is an acronym that helps us remember the steps to multiply two binomials correctly. It stands for:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms of the binomials.
• Last: Multiply the last terms of each binomial.

Let's apply this method to the expression $(x-7)(x-5)$.

1. First: Multiply the first terms of each binomial: $x * x = x^2$
2. Outer: Multiply the outer terms of the binomials: $x * -5 = -5x$
3. Inner: Multiply the inner terms of the binomials: $-7 * x = -7x$
4. Last: Multiply the last terms of each binomial: $-7 * -5 = 35$

Now, we combine these results:

$$x^2 - 5x - 7x + 35$$

This step is crucial as it ensures that we have accounted for all possible products between the terms of the two binomials. Each term in the first binomial must be multiplied by each term in the second binomial. The systematic nature of the FOIL method prevents us from missing any terms, which can lead to errors in the final simplified expression. The intermediate expression $x^2 - 5x - 7x + 35$ is a direct result of applying the FOIL method, and it sets the stage for the next step, which is combining like terms to simplify the expression further.

Combining Like Terms and Standard Form

The next step is to combine like terms in the expression $x^2 - 5x - 7x + 35$. Like terms are terms that have the same variable raised to the same power. In this case, $-5x$ and $-7x$ are like terms.

Combining these terms, we get:

$$-5x - 7x = -12x$$

So, the expression becomes:

$$x^2 - 12x + 35$$

This is the simplified form of the expression. Now, let's express it in standard form. The standard form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our case, the expression $x^2 - 12x + 35$ is already in standard form, with $a = 1$, $b = -12$, and $c = 35$.

Expressing the simplified expression in standard form is essential for several reasons. First, it provides a clear and organized representation of the polynomial, making it easier to identify the coefficients and the degree of the polynomial. Second, the standard form is crucial for further algebraic operations, such as solving quadratic equations, factoring polynomials, and graphing quadratic functions. The coefficients in the standard form ($a$, $b$, and $c$) play a significant role in determining the properties of the quadratic function, such as its vertex, axis of symmetry, and roots. Therefore, converting the simplified expression into standard form is not just a matter of convention but a practical step that facilitates further analysis and manipulation of the polynomial.

Table Representation of the FOIL Method

To further illustrate the FOIL method, we can represent the multiplication process in a table:

This table visually organizes the terms that need to be multiplied according to the FOIL method. It provides a clear overview of how each term in the first binomial is multiplied by each term in the second binomial, ensuring that no terms are missed. The table representation can be particularly helpful for students who are learning the FOIL method, as it breaks down the process into manageable steps and provides a visual aid for tracking the multiplications. Moreover, this table format can be extended to multiply polynomials with more terms, although the FOIL method itself is primarily designed for binomials. By using the table, we can systematically organize the terms and ensure that all possible products are accounted for, which is crucial for obtaining the correct simplified expression.

Resulting Simplified Expression

The resulting simplified expression after applying the FOIL method and combining like terms is:

$$x^2 - 12x + 35$$

This expression is in standard form, making it easy to analyze and use in further calculations. The process of simplifying expressions is a fundamental skill in algebra. By mastering techniques like the FOIL method, students can confidently manipulate polynomials, solve equations, and tackle more advanced mathematical concepts. The expression $x^2 - 12x + 35$ is a quadratic polynomial, and its standard form representation allows us to easily identify its coefficients and degree. The coefficient of the $x^2$ term is 1, the coefficient of the $x$ term is -12, and the constant term is 35. These coefficients play a crucial role in determining the properties of the quadratic function, such as its roots, vertex, and axis of symmetry. Furthermore, the standard form facilitates various algebraic manipulations, such as factoring the polynomial, completing the square, and applying the quadratic formula. Therefore, the ability to simplify expressions and express them in standard form is an essential skill for success in algebra and beyond.

Conclusion

In conclusion, we have successfully simplified the expression $(x-7)(x-5)$ using the FOIL method. The steps involved multiplying the First, Outer, Inner, and Last terms, and then combining like terms. The resulting expression in standard form is $x^2 - 12x + 35$. This process demonstrates the importance of the FOIL method in expanding binomials and the significance of expressing polynomials in standard form. Mastering these techniques is crucial for further studies in algebra and mathematics. The FOIL method provides a systematic approach to multiplying binomials, ensuring that all terms are accounted for and the simplification is accurate. The standard form, on the other hand, provides a clear and concise representation of the polynomial, making it easier to analyze and use in subsequent calculations. By understanding and applying these concepts, students can build a strong foundation in algebra and develop the problem-solving skills necessary for more advanced mathematical topics. Therefore, practice and proficiency in simplifying expressions using methods like FOIL and expressing the results in standard form are essential for success in mathematics.