Problem 50 - Elementary and Intermediate Algebra

Question

Perform the operations. See Example 4. $$ -2 h(7 h-2)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(-98h^3 + 56h^2 - 8h\).

Step 1: Understand the Expression

The expression given is \(-2h(7h - 2)^2\). We need to simplify this expression step-by-step, focusing first on the bracketed portion, \((7h - 2)^2\).

Step 2: Expand the Brackets

Square the expression inside the brackets: \((7h - 2)^2\). This means we apply: \((a-b)^2 = a^2 - 2ab + b^2\). So, we have:\((7h - 2)^2 = (7h)^2 - 2(7h)(2) + (2)^2 = 49h^2 - 28h + 4\).

Step 3: Distribute the -2h

We now distribute \(-2h\) across the expression we obtained in Step 2. Multiply each term in the expression \(49h^2 - 28h + 4\) by \(-2h\):1. \(-2h \times 49h^2 = -98h^3\)2. \(-2h \times (-28h) = 56h^2\)3. \(-2h \times 4 = -8h\)

Step 4: Combine the Results

Combine all the results from Step 3 to express the simplified form of the expression:\(-98h^3 + 56h^2 - 8h\).

Key Concepts

Polynomial Operations

Polynomial operations are a fundamental aspect of algebra. They involve performing arithmetic operations such as addition, subtraction, multiplication, and sometimes division on polynomial expressions. Polynomials are mathematical expressions consisting of variables, coefficients, and exponents that are combined using addition and subtraction.
Performing polynomial operations can involve the following:

• **Addition and Subtraction**: This requires combining like terms, which are terms that have the same variable raised to the same power. For example, in the polynomial expression \(3x^2 + 5x - x^2 - 4x\), you add and subtract like terms to simplify it as \((3x^2 - x^2) + (5x - 4x) = 2x^2 + x\).
• **Multiplication**: This involves using the distributive property, and sometimes the FOIL method, to multiply the terms across the polynomials. When working with a single variable, as seen in our given expression \(-2h(49h^2 - 28h + 4)\), you distribute \(-2h\) to each term within the polynomial to get \(-98h^3 + 56h^2 - 8h\).
• **Division**: Although not always encountered in basic polynomial operations, division can be performed using long division or synthetic division methods, particularly when dividing by a monomial or binomial.

Understanding polynomial operations is crucial as it forms the base for solving equations, factoring, and more advanced algebra topics. By mastering these operations, complex algebraic expressions become simpler and more manageable to solve.

Distributive Property

The distributive property is a key algebraic principle used to simplify expressions and solve equations, particularly when dealing with polynomials.
This property states that multiplying a sum by a factor is the same as multiplying each addend by the factor and then adding the products. It can be represented as:
\[a(b + c) = ab + ac\] In our example, once the expression \((7h-2)^2\) is expanded to \(49h^2 - 28h + 4\),the distributive property is used to multiply each term by \(-2h\),resulting in:

• \(-2h \times 49h^2 = -98h^3\)
• \(-2h \times (-28h) = 56h^2\)
• \(-2h \times 4 = -8h\)

The distributive property is particularly useful in algebra as it allows breaking down expressions into simpler parts that are easier to work with. It is essential, especially when simplifying expressions or solving equations that involve binomials or polynomials.

Exponents

Exponents are vital in algebra as they represent repeated multiplication of a base number. Understanding how to manipulate exponents is crucial for simplifying and solving algebraic expressions.
In expressions like \((7h - 2)^2\),the exponent affects not just the numbers, but also variables, requiring us to expand the expression by multiplying it by itself.

• **Exponent Laws**: Basic exponent laws help simplify expressions with exponents. For example, \((a^m)^n = a^{m \cdot n}\)and \(a^m \cdot a^n = a^{m+n}\).
• **Squaring a Binomial**: When squaring a binomial \((a-b)^2\),the formula \((a-b)^2 = a^2 - 2ab + b^2\)is used, allowing us to expand and simplify the expression.In our problem, applying this formula led to \(49h^2 - 28h + 4\).

Exponents allow mathematicians to express and manage large numbers and algebraic expressions concisely. Gaining proficiency in the utilization and properties of exponents is essential for tackling more complex mathematical challenges.