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Chapter 0: Problem 47

Perform the indicated operations and simplify. $$ (5 m-3)^{2} $$

Short Answer

Expert verified
25m^2 - 30m + 9

Step by step solution

01

Title - Expand the expression

The given expression is \( (5m - 3)^2 \). Use the formula \( (a - b)^2 = a^2 - 2ab + b^2 \) to expand it. Here, \( a = 5m \) and \( b = 3 \).
02

- Apply the formula

Substitute \( 5m \) for \( a \) and \(-3\) for \( b \) into the formula \((a - b)^2 = a^2 - 2ab + b^2 \): \[ (5m - 3)^2 = (5m)^2 - 2 \times 5m \times 3 + 3^2 \].
03

- Calculate each term

Calculate \( (5m)^2 = 25m^2 \), \(-2 \times 5m \times 3 = -30m \), and \( 3^2 = 9 \).
04

- Combine the terms

Add the calculated terms together: \( 25m^2 - 30m + 9 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial theorem
The binomial theorem is a cornerstone in algebra. It helps you expand expressions raised to a power, specifically binomials. A binomial is an algebraic expression with two terms. For example, \((a + b)^n\). The theorem provides a formula to expand such expressions without multiplying them manually.
The formula for the square of a binomial, used in this exercise, is: \((a - b)^2 = a^2 - 2ab + b^2\).
Understanding the binomial theorem helps simplify complex algebraic expressions. This is useful in probabilities, calculating compound interest, and much more.
Polynomial expressions
A polynomial is an expression consisting of variables and coefficients. It involves only non-negative integer exponents of variables. In this exercise, we transformed the binomial \((5m - 3)^2\) into its expanded polynomial form \((25m^2 - 30m + 9)\).
Here are key components of polynomials:
  • **Terms**: Parts of the expression separated by + or - signs.
  • **Coefficients**: Numbers in front of the variables (e.g., 25 in \(25m^2\)).
  • **Exponents**: Powers to which variables are raised (e.g., 2 in \(m^2\)).
Polynomials are widely applicable in fields like physics, engineering, and computer science.
Algebraic operations
Algebraic operations are the basic steps you perform to manipulate algebraic expressions. These include addition, subtraction, multiplication, division, and exponentiation of terms, variables, and constants.
In this particular exercise:
  • **Substitution**: Replacing variables with their values or other expressions.
  • **Applying formulas**: Using known identities like \((a - b)^2\) to simplify expressions.
  • **Combining like terms**: Simplifying the expression by adding or subtracting terms with the same variables raised to the same powers.
Understanding these operations is foundational in solving equations, graphing functions, and analyzing algebraic structures.

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Most popular questions from this chapter

Multiply and simplify. $$ (10 v-5)\left(\frac{1}{5} v^{2}-3 v+1\right) $$

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