Problem 224 Square each binomial using the B... [FREE SOLUTION]

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Chapter 5: Problem 224

Square each binomial using the Binomial Squares Pattern. $$ \left(3 x^{2}+2\right)^{2} $$

Short Answer

Expert verified

The expansion is $9x^4 + 12x^2 + 4$.

Step by step solution

Identify the Binomial

Recognize and write down the binomial expression given: $\left(3x^2 + 2\right)$.

Write Down the Binomial Squares Pattern

Recall the pattern for squaring a binomial: \[\left(a + b\right)^2 = a^2 + 2ab + b^2\]. In this case, $a = 3x^2$ and $b = 2$.

Calculate $a^2$

Square the first term $a$: \[\left(3x^2\right)^2 = 9x^4\].

Calculate $2ab$

Compute twice the product of the first term $a$ and the second term $b$: \[2 \times 3x^2 \times 2 = 12x^2\].

Calculate $b^2$

Square the second term $b$: \[2^2 = 4\].

Combine the Results

Add all the components together: \[9x^4 + 12x^2 + 4\].

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Key Concepts

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

squaring binomials

When we talk about squaring binomials, it refers to multiplying a binomial expression by itself.
This might seem tricky at first, but there鈥檚 a simple and consistent pattern we can use to simplify this process.
Let鈥檚 explore an example to understand it better: \[ \text{\textrm{(3x虏 + 2)虏}} \] This involves using the Binomial Squares Pattern. Let's break this pattern down further.
The formula for this pattern is: \[\text{\textrm{(a + b)虏 = a虏 + 2ab + b虏}} \] which helps you expand and simplify the expression without multiplying everything out individually.

polynomial expressions

Polynomial expressions consist of terms with variables raised to whole number exponents and their coefficients.
In our example, we have the polynomial \[\text{\textrm{3x虏 + 2}} \] Here's a step-by-step solution to squaring this binomial, which will ultimately result in another polynomial.
\[\text{\textrm{(3x虏)虏 = 9x鈦磢} \] Squaring the first term (3x虏).
Then, multiply the two terms and double it: \[\text{\textrm{2 脳 3x虏脳 2 = 12x虏}} \]
Lastly, square the second term: \[\text{\textrm{2虏 = 4}} \]
By combining these, we get our polynomial expression: \[\text{\textrm{9x鈦� + 12x虏 + 4}} \] Understanding this transformation is essential for mastering algebra.

algebraic multiplication

Algebraic multiplication is a fundamental skill in algebra. This includes multiplying polynomials, like when squaring binomials.
Here's a detailed look at the multiplication steps in our example.
When \[\text{\textrm{(a + b)虏}} \] is expanded, we perform three key multiplications: \[\text{\textrm{1. \ (3x虏)虏 to get 9x鈦磢} \] \[\text{\textrm{2. \ 2 脳 (3x虏) 脳 2 to get 12x虏}} \] \[\text{\textrm{3. \ 2虏 to get 4}} \] \ Each multiplication step builds on the previous one, combining like terms to form the final polynomial: \[\text{\textrm{9x鈦� + 12x虏 + 4}} \]
Understanding these steps ensures precise and accurate results in more complex algebraic problems.

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91影视

Square each binomial using the Binomial Squares Pattern. $$ \left(3 x^{2}+2\right)^{2} $$

Short Answer

Step by step solution

Identify the Binomial

Write Down the Binomial Squares Pattern

Calculate \(a^2\)

Calculate \(2ab\)

Calculate \(b^2\)

Combine the Results

Key Concepts

squaring binomials

polynomial expressions

algebraic multiplication

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