Problem 57 - Algebra 2

Question

REVIEW If \(g(x)=x^{2}+9 x+21\) and \(h(x)=2(x+5)^{2},\) which is an equivalent form of \(h(x)-g(x) ?\) $$ \begin{array}{l}{\mathbf{F}-x^{2}-11 x-29} \\ {\mathbf{G} x^{2}+11 x+29} \\\ {\mathbf{H} x+4} \\ {\mathbf{J} \quad x^{2}+7 x+11}\end{array} $$

Step-by-Step Solution

Answer

Option G: \(x^2 + 11x + 29\).

Step 1: Expand and Simplify h(x)

Given that \(h(x) = 2(x+5)^2\), first expand \((x+5)^2\). This results in: \((x+5)^2 = x^2 + 10x + 25\). Now, distribute the 2: \(h(x) = 2(x^2 + 10x + 25) = 2x^2 + 20x + 50\).

Step 2: Calculate h(x) - g(x)

Subtract \(g(x) = x^2 + 9x + 21\) from the expanded form of \(h(x)\). Calculate: \((2x^2 + 20x + 50) - (x^2 + 9x + 21)\).

Step 3: Combine Like Terms

Start by combining like terms: - \(2x^2 - x^2 = x^2\)- \(20x - 9x = 11x\)- \(50 - 21 = 29\)Thus, \(h(x) - g(x) = x^2 + 11x + 29\).

Step 4: Select the Correct Answer Option

Compare the solved expression \(x^2 + 11x + 29\) with the given options. The correct choice is Option G: \(x^2 + 11x + 29\).

Key Concepts

Equivalent Forms

In algebra, equivalent forms are different expressions that represent the same polynomial. When we transform expressions by processes like expanding, factoring, or simplifying, we're often seeking an equivalent form. This concept allows us to take complex polynomial forms and express them in simpler or more useful ways. Let's look at how we derived the equivalent form of the polynomial function in the exercise.
To find an equivalent form of \( h(x) - g(x) \), the polynomial expressions must first be simplified and expanded. The simplification leads us to a form that, although it looks different, expresses the same mathematical idea as the original functions. For example, in the exercise, \( h(x) = 2(x+5)^2 \) was expanded to \( 2x^2 + 20x + 50 \). Both expressions are equivalent because they produce the same result for any value of \( x \).

• Start by transforming complex expressions into their simplified equivalent forms.
• Ensure all steps of expansion and simplification maintain mathematical equality.
• Compare the simplified expression with given options to find the equivalent form.

Understanding equivalent forms is crucial in algebra, as it allows for flexible manipulation of polynomial expressions in various contexts.

Polynomial Subtraction

Polynomial subtraction involves removing the terms of one polynomial from another. It's a crucial operation that helps handle expressions involving multiple polynomial terms. In the exercise, we had to subtract \( g(x) \) from the expanded \( h(x) \), written as \( (2x^2 + 20x + 50) - (x^2 + 9x + 21) \).
Here's a simple way to approach polynomial subtraction:

• Align like terms, which are terms with the same degree.
• Subtract the coefficients of the like terms.
• Combine the results to form the new polynomial.

In this exercise, we did the following to subtract the polynomials:
- For \( x^2 \) terms: \( 2x^2 - x^2 = x^2 \)
- For \( x \) terms: \( 20x - 9x = 11x \)
- For constant terms: \( 50 - 21 = 29 \)
This operation results in the new polynomial \( x^2 + 11x + 29 \), showing how crucial it is to manage terms carefully during polynomial subtraction.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. Understanding how to handle them is essential in solving algebra problems. In our exercise, both \( g(x) = x^2 + 9x + 21 \) and \( h(x) = 2(x+5)^2 \) are algebraic expressions that characterize polynomial functions.
An effective approach to handling algebraic expressions involves:

• Identifying constants \( (21, 50) \) and terms with variables \( (x, x^2) \).
• Simplifying expressions to their most compact forms, such as through distribution or combining like terms.
• Recognizing consistent structure and using rules to expand or factor expressions.

For example, expanding \( (x+5)^2 \) needed knowledge of the binomial theorem and distribution of the coefficient 2 over each term. This led us to combine and simplify terms.
Algebraic expressions form the foundation of algebra and problem-solving. Mastery of their manipulation allows for solving more complex equations and understanding the core ideas in algebraic thinking.