Factor The Expression:1. X 2 + Cos ⁡ X + 21 X^2 + \cos X + 21 X 2 + Cos X + 21

Introduction

In mathematics, factoring an expression is a fundamental concept that involves expressing a given expression as a product of simpler expressions. This technique is widely used in algebra, trigonometry, and other branches of mathematics to simplify complex expressions and solve equations. In this article, we will focus on factoring an expression that involves a quadratic term and a trigonometric function, specifically the cosine function.

The Expression to be Factored

The given expression is $x^2 + \cos x + 21$. This expression consists of a quadratic term $x^2$, a trigonometric term $\cos x$, and a constant term $21$. Our goal is to factor this expression into simpler terms.

Understanding the Quadratic Formula

Before we proceed with factoring the expression, let's recall the quadratic formula, which is a fundamental concept in algebra. The quadratic formula states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

This formula can be used to solve quadratic equations, but it is not directly applicable to our expression, which involves a trigonometric term.

Factoring the Expression

To factor the expression $x^2 + \cos x + 21$, we need to find two numbers whose product is $21$ and whose sum is $1$. However, since the expression involves a trigonometric term, we cannot simply use the quadratic formula or factorization techniques that rely on the sum and product of coefficients.

Instead, we can use a trigonometric identity to rewrite the expression in a form that is easier to factor. Specifically, we can use the identity $\cos^2 x + 1 = \sec^2 x$ to rewrite the expression as:

$$x^2 + \cos x + 21 = x^2 + \cos x + 21 + \cos^2 x - \cos^2 x$$

This allows us to rewrite the expression as:

$$x^2 + \cos x + 21 = (x^2 + \cos x + 21 + \cos^2 x) - \cos^2 x$$

Now, we can factor the expression as:

$$x^2 + \cos x + 21 = (x^2 + \cos x + 21 + \cos^2 x) - \cos^2 x = (x^2 + \cos x + 21)(1 - \cos x)$$

However, this factorization is not correct, as it does not take into account the fact that the expression involves a trigonometric term.

A Correct Approach to Factoring the Expression

To factor the expression $x^2 + \cos x + 21$, we need to use a different approach. Specifically, we can use the fact that the expression can be written as a sum of two squares:

$$x^2 + \cos x + 21 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4}\cos^2 x$$

This allows us to rewrite the expression as:

$$x^2 + \cos x + 21 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4}\cos^2 x = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4}(1 - \sin^2 x)$$

Now, we can simplify the expression further by using the identity $\sin^2 x + \cos^2 x = 1$:

$$x^2 + \cos x + 21 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4}(1 - \sin^2 x) = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}\sin^2 x$$

This allows us to rewrite the expression as:

$$x^2 + \cos x + 21 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}\sin^2 x = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}\sin^2 x = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2$$

Now, we can factor the expression as:

x^2 + \cos x + 21 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{4}(\sin x)^2 = (x + \frac{1}{2}\cos x)^2 + \frac{21}{4} - \frac{1}{4} + \frac{1}{<br/> **Q&A: Factoring the Expression $x^2 + \cos x + 21$** =====================================================

Q: What is the main goal of factoring the expression $x^2 + \cos x + 21$?

A: The main goal of factoring the expression $x^2 + \cos x + 21$ is to express it as a product of simpler expressions, which can help us simplify complex expressions and solve equations.

Q: Why is factoring the expression $x^2 + \cos x + 21$ important?

A: Factoring the expression $x^2 + \cos x + 21$ is important because it allows us to simplify complex expressions and solve equations. This is particularly useful in mathematics, where complex expressions are often encountered.

Q: What are some common techniques used to factor expressions?

A: Some common techniques used to factor expressions include:

• Quadratic formula: This technique is used to solve quadratic equations of the form $ax^2 + bx + c = 0$.
• Factoring by grouping: This technique involves grouping terms in an expression and factoring out common factors.
• Factoring by difference of squares: This technique involves factoring expressions of the form $a^2 - b^2$.

Q: How can we use trigonometric identities to factor expressions?

A: We can use trigonometric identities to factor expressions by rewriting the expression in a form that is easier to factor. For example, we can use the identity $\cos^2 x + 1 = \sec^2 x$ to rewrite the expression $x^2 + \cos x + 21$ as $(x^2 + \cos x + 21 + \cos^2 x) - \cos^2 x$.

Q: What are some common mistakes to avoid when factoring expressions?

A: Some common mistakes to avoid when factoring expressions include:

• Not checking for common factors: Make sure to check for common factors in the expression before attempting to factor it.
• Not using trigonometric identities: Trigonometric identities can be useful in factoring expressions, so make sure to use them when necessary.
• Not simplifying the expression: Make sure to simplify the expression after factoring it to ensure that it is in its simplest form.

Q: How can we use the factored form of an expression to solve equations?

A: We can use the factored form of an expression to solve equations by setting each factor equal to zero and solving for the variable. For example, if we have the factored form $(x + 1)(x - 2) = 0$, we can set each factor equal to zero and solve for $x$ to get $x + 1 = 0$ and $x - 2 = 0$.

Q: What are some real-world applications of factoring expressions?

A: Factoring expressions has many real-world applications, including:

• Engineering: Factoring expressions is used in engineering to simplify complex equations and solve problems.
• Physics: Factoring expressions is used in physics to simplify complex equations and solve problems related to motion and energy.
• Computer science: Factoring expressions is used in computer science to simplify complex algorithms and solve problems related to data analysis and machine learning.

Conclusion

Factoring the expression $x^2 + \cos x + 21$ is an important concept in mathematics that has many real-world applications. By understanding the techniques used to factor expressions and avoiding common mistakes, we can simplify complex expressions and solve equations.