Problem 24 - Elementary Algebra

Question

Multiply. $$ (5 x 2-7 x+9)(-5) $$

Step-by-Step Solution

Answer

The expression multiplies to \(-25x^2 + 35x - 45\).

Step 1: Understand the Expression

The expression \((5x^2 - 7x + 9)(-5)\) involves multiplying a polynomial by a constant, which is \(-5\). Our task is to distribute \(-5\) to each term in the polynomial \(5x^2 - 7x + 9\).

Step 2: Distribute -5 to Each Term

Distribute \(-5\) to each term inside the parentheses:1. Multiply \(-5\) by \(5x^2\):\[-5 imes 5x^2 = -25x^2\]2. Multiply \(-5\) by \(-7x\):\[-5 imes (-7x) = 35x\]3. Multiply \(-5\) by \(9\):\[-5 imes 9 = -45\]

Step 3: Combine the Resulting Terms

After distributing, combine the terms to form the expanded expression:The terms are \(-25x^2 + 35x - 45\).

Key Concepts

Distributive Property

The distributive property is an essential rule in algebra that allows for easier multiplication of expressions. In the exercise, we are asked to distribute the constant \(-5\) across the polynomial \(5x^2 - 7x + 9\). This means that \(-5\) is multiplied with each term within the polynomial separately.
This concept can be particularly useful when working with algebraic expressions because it enables you to break down complex equations into simpler parts.
How Does It Work?
Think of the distributive property as a way of distributing the constant across the terms in the expression:

• Multiply \(-5\) by the first term \(5x^2\) to get \(-25x^2\).
• Then, multiply \(-5\) by the second term \(-7x\), which results in \(+35x\).
• Finally, multiply \(-5\) by the constant \(9\) to get \(-45\).

By applying the distributive property, you can convert a polynomial multiplication problem into individual simpler multiplication tasks.

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra that involve numbers, variables, and operations like addition, subtraction, and multiplication. In the given exercise \(5x^2 - 7x + 9\), the expression consists of different terms that are crucial to understand:
Keys of Algebraic Expressions

• The term \(5x^2\) is a "quadratic term," where "x" is the variable and "2" is its exponent.
• The term \(-7x\) is a "linear term," with the variable "x" raised to the power of 1.
• The constant term \(9\) has no variable attached to it.

It's helpful to recognize each part of an algebraic expression to easily apply operations, like distribution in this case, successfully.
Understanding how variables and coefficients interact is key to simplifying and solving algebraic problems.

Constant Multiplication

Constant multiplication refers to multiplying a constant with each term in an algebraic expression, just like in our example. This process doesn't change the algebraic structure; instead, it scales each term by the factor of the constant.
Steps in Constant Multiplication

• First, recognize that the constant \(-5\) needs to be distributed to every term within the polynomial.
• Then, execute the multiplication for each term:
  • \(-5 \times 5x^2 = -25x^2\)
  • \(-5 \times (-7x) = 35x\)
  • \(-5 \times 9 = -45\)

These calculations result in a new expression that is systematically derived by applying the constant to every part of the polynomial.
This technique is crucial for transforming and simplifying expressions, and mastering it equips you to handle various algebraic equations more effectively.