Factor The Expression $u^2 + 10u + 25$.

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Introduction

Factoring an expression is a fundamental concept in algebra that involves expressing a given polynomial as a product of simpler polynomials. In this article, we will focus on factoring the expression $u^2 + 10u + 25$. This expression is a quadratic expression, and factoring it will help us understand its roots and behavior.

Understanding the Expression

The given expression is a quadratic expression in the form of $ax^2 + bx + c$. In this case, the coefficients are $a = 1$, $b = 10$, and $c = 25$. To factor this expression, we need to find two numbers whose product is $ac$ and whose sum is $b$.

The Factoring Process

To factor the expression $u^2 + 10u + 25$, we need to find two numbers whose product is $1 \times 25 = 25$ and whose sum is $10$. These numbers are $5$ and $5$, since $5 \times 5 = 25$ and $5 + 5 = 10$.

Factoring the Expression

Using the numbers $5$ and $5$, we can rewrite the expression as:

u2+10u+25=(u+5)(u+5)u^2 + 10u + 25 = (u + 5)(u + 5)

This is the factored form of the expression.

Simplifying the Factored Form

We can simplify the factored form by combining the two identical binomials:

(u+5)(u+5)=(u+5)2(u + 5)(u + 5) = (u + 5)^2

This is the simplified factored form of the expression.

Understanding the Roots

The factored form of the expression $(u + 5)^2$ tells us that the expression has a repeated root at $u = -5$. This means that the expression has a double root at $u = -5$.

Graphical Representation

The graph of the expression $u^2 + 10u + 25$ is a parabola that opens upward. The factored form $(u + 5)^2$ tells us that the graph has a repeated root at $u = -5$, which means that the graph touches the x-axis at $u = -5$.

Real-World Applications

Factoring expressions has many real-world applications in fields such as physics, engineering, and economics. For example, factoring expressions can help us understand the behavior of physical systems, such as the motion of objects under the influence of gravity.

Conclusion

In conclusion, factoring the expression $u^2 + 10u + 25$ involves finding two numbers whose product is $ac$ and whose sum is $b$. The factored form of the expression is $(u + 5)^2$, which tells us that the expression has a repeated root at $u = -5$. Factoring expressions has many real-world applications and is an essential tool in algebra.

Additional Tips and Tricks

  • When factoring expressions, it's essential to look for common factors and to use the distributive property to simplify the expression.
  • Factoring expressions can help us understand the behavior of physical systems and can be used to solve problems in fields such as physics and engineering.
  • The factored form of an expression can be used to find the roots of the expression, which can be used to solve problems in fields such as economics and finance.

Common Mistakes to Avoid

  • When factoring expressions, it's essential to look for common factors and to use the distributive property to simplify the expression.
  • Factoring expressions can be challenging, especially when dealing with complex expressions. It's essential to take your time and to use the distributive property to simplify the expression.
  • The factored form of an expression can be used to find the roots of the expression, which can be used to solve problems in fields such as economics and finance.

Conclusion

In conclusion, factoring the expression $u^2 + 10u + 25$ involves finding two numbers whose product is $ac$ and whose sum is $b$. The factored form of the expression is $(u + 5)^2$, which tells us that the expression has a repeated root at $u = -5$. Factoring expressions has many real-world applications and is an essential tool in algebra.

Final Thoughts

Factoring expressions is a fundamental concept in algebra that involves expressing a given polynomial as a product of simpler polynomials. The factored form of an expression can be used to find the roots of the expression, which can be used to solve problems in fields such as economics and finance. By understanding the factored form of an expression, we can gain a deeper understanding of the behavior of physical systems and can use this knowledge to solve problems in fields such as physics and engineering.

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Introduction

Factoring expressions is a fundamental concept in algebra that involves expressing a given polynomial as a product of simpler polynomials. In this article, we will provide a Q&A guide to help you understand the concept of factoring expressions and how to apply it to solve problems.

Q: What is factoring an expression?

A: Factoring an expression involves expressing a given polynomial as a product of simpler polynomials. This means that we need to find the factors of the expression and multiply them together to get the original expression.

Q: Why is factoring an expression important?

A: Factoring an expression is important because it helps us understand the behavior of the expression and its roots. By factoring an expression, we can find the roots of the expression, which can be used to solve problems in fields such as physics, engineering, and economics.

Q: How do I factor an expression?

A: To factor an expression, we need to look for common factors and use the distributive property to simplify the expression. We can also use the factoring formulas, such as the difference of squares formula and the sum of cubes formula, to factor the expression.

Q: What are the common factoring formulas?

A: The common factoring formulas are:

  • Difference of squares formula: a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)
  • Sum of cubes formula: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)
  • Difference of cubes formula: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Q: How do I use the distributive property to simplify an expression?

A: To use the distributive property to simplify an expression, we need to multiply each term in the expression by the factor. For example, if we have the expression 2x+32x + 3, we can multiply each term by the factor xx to get 2x2+3x2x^2 + 3x.

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

A: The common mistakes to avoid when factoring expressions are:

  • Not looking for common factors
  • Not using the distributive property to simplify the expression
  • Not using the factoring formulas to factor the expression

Q: How do I find the roots of an expression?

A: To find the roots of an expression, we need to set the expression equal to zero and solve for the variable. For example, if we have the expression x2+4x+4=0x^2 + 4x + 4 = 0, we can set it equal to zero and solve for xx to get x=−2x = -2.

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

A: The real-world applications of factoring expressions are:

  • Physics: Factoring expressions can help us understand the behavior of physical systems, such as the motion of objects under the influence of gravity.
  • Engineering: Factoring expressions can help us design and optimize systems, such as bridges and buildings.
  • Economics: Factoring expressions can help us understand the behavior of economic systems, such as the behavior of supply and demand.

Conclusion

In conclusion, factoring expressions is a fundamental concept in algebra that involves expressing a given polynomial as a product of simpler polynomials. By understanding the concept of factoring expressions and how to apply it to solve problems, we can gain a deeper understanding of the behavior of physical systems and can use this knowledge to solve problems in fields such as physics, engineering, and economics.

Final Thoughts

Factoring expressions is a powerful tool that can be used to solve problems in a variety of fields. By understanding the concept of factoring expressions and how to apply it to solve problems, we can gain a deeper understanding of the behavior of physical systems and can use this knowledge to solve problems in fields such as physics, engineering, and economics.

Additional Resources

  • Khan Academy: Factoring Expressions
  • Mathway: Factoring Expressions
  • Wolfram Alpha: Factoring Expressions

Common Questions

  • Q: What is the difference between factoring and simplifying an expression?
  • A: Factoring an expression involves expressing a given polynomial as a product of simpler polynomials, while simplifying an expression involves combining like terms and eliminating any unnecessary terms.
  • Q: How do I factor a quadratic expression?
  • A: To factor a quadratic expression, we need to look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
  • Q: What are the common factoring formulas?
  • A: The common factoring formulas are the difference of squares formula, the sum of cubes formula, and the difference of cubes formula.