Problem 48 Find each product. $$\left(5 x... [FREE SOLUTION]

Chapter 0: Problem 48

Find each product. $$\left(5 x^{2}-3\right)^{2}$$

Short Answer

Expert verified

The expanded form of $(5x^2 - 3)^2$ is $25x^4 - 30x^2 + 9$.

Step by step solution

Identify $a$ and $b$ in the binomial

In the binomial $(5x^2 - 3)^2$, identify $a = 5x^2$ and $b = 3$. Remember, the formula for the square of a binomial is $a^2 - 2ab + b^2$.

Substitute $a$ and $b$ into the formula

Substitute $a = 5x^2$ and $b = 3$ into the formula, resulting in $(5x^2)^2 - 2 * 5x^2 * 3 + 3^2$.

Simplify the expression

Simplify the expression $(5x^2)^2 - 2 * 5x^2 * 3 + 3^2$ to $25x^4 - 30x^2 + 9$

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Key Concepts

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

Squaring Binomials

Squaring a binomial involves expanding the expression $a - b$^2 into its full polynomial form. This is not the same as just squaring each term individually. Instead, it follows a specific algebraic identity: $(a - b)^2 = a^2 - 2ab + b^2$.
The formula shows the result of multiplying a binomial by itself. For $5x^2 - 3$, squaring involves:

Finding the square of the first term, which is $ (5x^2)^2 = 25x^4$.
Calculating twice the product of both terms, $-2 \times 5x^2 \times 3 = -30x^2$.
Squaring the last term, $(3)^2 = 9$.

These components are then combined into a single polynomial expression.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents. The expression \25x^4 - 30x^2 + 9\ is a polynomial resulting from squaring the binomial $5x^2 - 3$.
Polynomial terms are composed of:

A coefficient, like 25, which multiplies the variable term $x^4$.
A variable raised to an exponent, such as $x^4$ and $x^2$ in our expression.

There are different degrees of polynomials based on the highest power of the variable. Here, the highest power, $x^4$, makes it a quartic polynomial.
Each term's contribution is important in shaping the polynomial's curve on a graph.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging, simplifying, or rewriting expressions using the rules of algebra to find a solution or simplify results. In the context of our problem, algebraic manipulation is used in the simplification process.
To simplify the expression $25x^4 - 30x^2 + 9$:

Recognize and rearrange the terms according to standard practice, which often involves listing in decreasing order of the exponents of the variable.
Simplify components sequentially: evaluate each term through operations like distributing products or factoring.

By applying algebraic manipulation skills, you can convey complex expressions clearly and make problem-solving more approachable.

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91影视

Find each product. $$\left(5 x^{2}-3\right)^{2}$$

Short Answer

Step by step solution

Identify \(a\) and \(b\) in the binomial

Substitute \(a\) and \(b\) into the formula

Simplify the expression

Key Concepts

Squaring Binomials

Polynomial Expressions

Algebraic Manipulation

One App. One Place for Learning.

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