Problem 84 - Elementary Algebra

Question

\(m(9 m-2)-7(4 m-5)\)

Step-by-Step Solution

Answer

9m^2 - 30m + 35

Step 1: Expand the expression

Distribute the multiplication over the terms inside the parentheses. Multiply each term in the parentheses with the term outside it.For the first part: \[ m(9m - 2) = 9m^2 - 2m \]For the second part: \[ -7(4m - 5) = -28m + 35 \]

Step 2: Combine like terms

Add the results from the expanded expression together: \[ 9m^2 - 2m - 28m + 35 \]Combine the terms with the same powers of \(m\) (like terms).\[ 9m^2 + (-2m - 28m) + 35 \]\[ 9m^2 - 30m + 35 \]

Key Concepts

Distribution

In algebra, distribution is a crucial property that helps break down complex expressions. It's based on the distributive property, which states that multiplying a number (or variable) by a group of numbers added together is the same as doing each multiplication separately.

For example, in the expression \(m(9m - 2)\), we distribute 'm' into each term inside the parentheses:

• \(m \times 9m = 9m^2\)
• \(m \times -2 = -2m\)

Doing the same for \(-7(4m - 5)\), we get:

• \(-7 \times 4m = -28m\)
• \(-7 \times -5 = 35\)

Remember: distribute carefully and watch for negative signs!

Like Terms

Combining like terms is another essential concept in algebra. Like terms are terms that have the same variable raised to the same power. For instance, \( -2m \) and \(-28m\) are like terms because they both have the variable 'm' raised to the first power.

When we combine like terms, we simply add or subtract their coefficients. In our given expression:
After expanding, we get \(9m^2 - 2m - 28m + 35\).

• Combine \(-2m\) and \(-28m\) to get \(-30m\).

This gives the simplified expression: \[9m^2 - 30m + 35\].
Remember: Always combine terms with matching variables and exponents!

Polynomials

A polynomial is a mathematical expression made up of variables, exponents, and coefficients, added or subtracted together. The expression in our example, \(m(9m - 2) - 7(4m - 5)\), is a polynomial after expanding and simplifying it.

This expression becomes:
\[9m^2 - 30m + 35\].
It contains:

• A quadratic term \(9m^2\) (second-degree term)
• A linear term \(-30m\) (first-degree term)
• A constant term \(35\) (zero-degree term)

Remember: Polynomials can have multiple terms and each term's degree depends on the exponent!

Expanding Expressions

Expanding expressions is the process of using distribution to remove parentheses and simplify. In the given problem:
First, distribute the terms:

• \(m(9m - 2) = 9m^2 - 2m\)
• \(-7(4m - 5) = -28m + 35\)

Next, combine all the terms together:
\[9m^2 - 2m - 28m + 35\].
Finishing with combining like terms to simplify:
\[9m^2 - 30m + 35\].
Remember: Expanding makes complex expressions simpler and easier to solve!