Problem 5 - Elementary Algebra

Question

\(m=5 p^{3}-2 p+7\). Determine the value of \(m\) if \(p=-2\).

Step-by-Step Solution

Answer

Answer: When \(p=-2\), the value of \(m\) is \(-36\).

Step 1: Substitute the value of p in the expression

Replace the variable \(p\) with the given value \(-2\) in the expression \(m=5 p^{3}-2 p+7\), so the expression becomes: \(m=5(-2)^3 - 2(-2) + 7\).

Step 2: Evaluate powers

Calculate \((-2)^3\) and replace it in the expression, which gives: \(m=5(-8) - 2(-2) + 7\).

Step 3: Multiply the coefficients by the terms

Multiply \(5\) with \(-8\) and \(-2\) with \(-2\), and then substitute the results into the expression: \(m=-40 + 4 + 7\).

Step 4: Add and subtract the terms

Combine the terms \(-40\), \(4\), and \(7\), so the expression becomes: \(m=-40 + 4 + 7\), which simplifies to: \(m=-36\). The value of \(m\) when \(p=-2\) is \(-36\).

Key Concepts

Evaluating Algebraic Expressions

The process of finding the value of algebraic expressions by replacing variables with numbers is known as evaluating algebraic expressions. It is a fundamental skill in algebra that allows us to discover values for particular situations. For example, in the exercise provided, the expression is given as m=5p^3-2p+7 and we need to find the value of m when p is -2. The first important step is to substitute -2 for every occurrence of p in the expression.

Keep in mind that the order of operations matters significantly here, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). After substitution, we proceed by following this order. This ensures that the powers and products are evaluated before any addition or subtraction takes place.

Understanding variable substitution and the correct order of operations are vital for correctly evaluating algebraic expressions. By mastering these concepts, students will find it easier to handle more complex mathematical problems.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and sometimes division of polynomials. In our example, we're dealing with a polynomial in the form of a cubic expression, represented by 5p^3-2p+7. Operations on polynomials follow the same arithmetic rules that apply to integers, with the addition and subtraction concerning like terms -- terms that have the same variable raised to the same power.

When substituting a value into a polynomial, it’s essential to carefully perform polynomial operations to ensure the accuracy of the result. 5(-2)^3 represents the multiplication operation, where we multiply the coefficient 5 by the cube of -2. This step illustrates not just substitution, but also an understanding of how to manipulate polynomial terms.

Tip for Success:

Always double-check that you’ve combined like terms properly and have not missed any terms during your calculations. Using a systematic approach to tackle each part of the polynomial helps avoid errors.

Powers and Exponents

Powers and exponents play a crucial role when working with algebraic expressions, especially in polynomials like the one in our exercise. The term p^3 signifies p raised to the power of 3, also read as 'p cubed'. It is the same as multiplying p by itself three times: p*p*p. When evaluating, it’s critical to remember that the sign of the number is also raised to the power, which is why (-2)^3 becomes -8, not +8.

Understanding the rules governing powers and exponents, such as a^n * a^m = a^(n+m) and (a^n)^m = a^(n*m), help simplify complicated algebraic expressions and facilitate their evaluation. For negative bases, the power's parity will affect the sign of the result, which is an important detail to remember, as it impacts all subsequent arithmetic operations.

Practice Exercise:

Try evaluating expressions with different powers, like (-3)^2 or 4^3, to gain a better intuition for how powers of negative and positive numbers work.