3. Expand The Following Expressions:a) $y(y-4$\]b) $r(r+5$\]c) $x(2x-5$\]d) $q(-4q+8$\]e) $z(-3z+2$\]f) $m(-m-5$\]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and expanding them is a crucial skill to master. In this article, we will explore the process of expanding various algebraic expressions, including quadratic and linear expressions. We will also provide examples and explanations to help you understand the concept better.

Expanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a squared variable. The general form of a quadratic expression is ax^2 + bx + c, where a, b, and c are constants, and x is the variable.

Expanding y(y−4)y(y-4)

To expand the expression y(y−4)y(y-4), we need to multiply the two binomials using the distributive property.

y(y-4) = y \cdot y - y \cdot 4
= y^2 - 4y

As you can see, the expanded expression is a quadratic expression in the form of y^2 - 4y.

Expanding r(r+5)r(r+5)

To expand the expression r(r+5)r(r+5), we need to multiply the two binomials using the distributive property.

r(r+5) = r \cdot r + r \cdot 5
= r^2 + 5r

The expanded expression is a quadratic expression in the form of r^2 + 5r.

Expanding x(2x−5)x(2x-5)

To expand the expression x(2x−5)x(2x-5), we need to multiply the two binomials using the distributive property.

x(2x-5) = x \cdot 2x - x \cdot 5
= 2x^2 - 5x

The expanded expression is a quadratic expression in the form of 2x^2 - 5x.

Expanding q(−4q+8)q(-4q+8)

To expand the expression q(−4q+8)q(-4q+8), we need to multiply the two binomials using the distributive property.

q(-4q+8) = q \cdot (-4q) + q \cdot 8
= -4q^2 + 8q

The expanded expression is a quadratic expression in the form of -4q^2 + 8q.

Expanding z(−3z+2)z(-3z+2)

To expand the expression z(−3z+2)z(-3z+2), we need to multiply the two binomials using the distributive property.

z(-3z+2) = z \cdot (-3z) + z \cdot 2
= -3z^2 + 2z

The expanded expression is a quadratic expression in the form of -3z^2 + 2z.

Expanding m(−m−5)m(-m-5)

To expand the expression m(−m−5)m(-m-5), we need to multiply the two binomials using the distributive property.

m(-m-5) = m \cdot (-m) + m \cdot (-5)
= -m^2 - 5m

The expanded expression is a quadratic expression in the form of -m^2 - 5m.

Conclusion

Expanding algebraic expressions is a crucial skill in mathematics, and it requires a good understanding of the distributive property. In this article, we have explored the process of expanding various algebraic expressions, including quadratic and linear expressions. We have also provided examples and explanations to help you understand the concept better. With practice and patience, you can master the art of expanding algebraic expressions and become proficient in solving mathematical problems.

Tips and Tricks

  • Always use the distributive property to expand algebraic expressions.
  • Make sure to multiply each term in the first binomial by each term in the second binomial.
  • Simplify the expression by combining like terms.
  • Use parentheses to group terms and make the expression easier to read.

Practice Problems

  • Expand the expression x(x+2)x(x+2).
  • Expand the expression y(y−3)y(y-3).
  • Expand the expression z(z+1)z(z+1).
  • Expand the expression m(m−2)m(m-2).
  • Expand the expression q(q+4)q(q+4).

Answer Key

  • x(x+2)=x2+2xx(x+2) = x^2 + 2x
  • y(y−3)=y2−3yy(y-3) = y^2 - 3y
  • z(z+1)=z2+zz(z+1) = z^2 + z
  • m(m−2)=m2−2mm(m-2) = m^2 - 2m
  • q(q+4)=q2+4qq(q+4) = q^2 + 4q
    Frequently Asked Questions: Expanding Algebraic Expressions ===========================================================

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to expand algebraic expressions by multiplying each term in the first binomial by each term in the second binomial.

Q: How do I expand an algebraic expression using the distributive property?

A: To expand an algebraic expression using the distributive property, you need to multiply each term in the first binomial by each term in the second binomial. For example, to expand the expression x(x+2)x(x+2), you would multiply xx by xx and xx by 22, resulting in x2+2xx^2 + 2x.

Q: What is the difference between expanding and simplifying an algebraic expression?

A: Expanding an algebraic expression involves multiplying each term in the first binomial by each term in the second binomial, while simplifying an algebraic expression involves combining like terms to make the expression easier to read.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, to simplify the expression x2+2x+3xx^2 + 2x + 3x, you would combine the like terms 2x2x and 3x3x to get 5x5x, resulting in the simplified expression x2+5xx^2 + 5x.

Q: What is the importance of expanding and simplifying algebraic expressions?

A: Expanding and simplifying algebraic expressions is important because it allows us to solve mathematical problems more easily. By expanding and simplifying algebraic expressions, we can make the expressions easier to read and understand, which can help us to solve problems more efficiently.

Q: How do I know when to expand and simplify an algebraic expression?

A: You should expand and simplify an algebraic expression when you need to solve a mathematical problem that involves the expression. For example, if you are given the expression x(x+2)x(x+2) and you need to find the value of xx, you would expand the expression to get x2+2xx^2 + 2x and then simplify it to make it easier to solve.

Q: What are some common mistakes to avoid when expanding and simplifying algebraic expressions?

A: Some common mistakes to avoid when expanding and simplifying algebraic expressions include:

  • Not using the distributive property to expand expressions
  • Not combining like terms when simplifying expressions
  • Not checking for errors when expanding and simplifying expressions

Q: How can I practice expanding and simplifying algebraic expressions?

A: You can practice expanding and simplifying algebraic expressions by working on problems and exercises that involve these concepts. You can also use online resources and tools to help you practice and improve your skills.

Q: What are some real-world applications of expanding and simplifying algebraic expressions?

A: Expanding and simplifying algebraic expressions has many real-world applications, including:

  • Solving mathematical problems in science, technology, engineering, and mathematics (STEM) fields
  • Modeling real-world situations using algebraic expressions
  • Solving optimization problems in business and economics

Conclusion

Expanding and simplifying algebraic expressions is an important skill in mathematics that has many real-world applications. By understanding the distributive property and how to expand and simplify algebraic expressions, you can solve mathematical problems more efficiently and effectively. Remember to practice regularly and avoid common mistakes to improve your skills.