Problem 88 Factor. $$(2 a+3)^{2}-25 b^{2}... [FREE SOLUTION]

Chapter 10: Problem 88

Factor. $$(2 a+3)^{2}-25 b^{2}$$

Short Answer

Expert verified

The factored form of the expression $(2a+3)^2 - 25b^2$ is $(2a+3+5b)(2a+3-5b)$

Step by step solution

Identify the Squares

In the expression $(2a+3)^2 - 25b^2$, the first square is $(2a+3)^2$ and the second square is $25b^2$. Therefore, we can set $a = (2a+3)$ and $b = 5b$.

Apply the Difference of Squares Formula

Now we apply the difference of squares formula to our expression. According to the formula, we can rewrite the expression as follows: $(2a+3+5b)(2a+3-5b)$

Simplify the Expression

The final simplified form is: $(2a+3+5b)(2a+3-5b)$

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Key Concepts

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

Difference of Squares

When you encounter an expression like - $(2a+3)^2 - 25b^2$, which is a type of polynomial, you can factor it using the difference of squares method. This method involves expressions that can be written in the form $a^2 - b^2$, which factors to $(a + b)(a - b)$. The key steps are to identify the two squares involved.

For example, in the expression:

$(2a+3)^2$, the first square is represented by $ (2a+3) $.
The second square is $25b^2$, represented by $5b$.

With these identifications, the difference of squares formula can be applied directly to achieve factoring.

This method is particularly useful because it turns complex expressions into simpler binomial multiplication. Recognizing and remembering this structure of subtraction between two perfect squares is crucial.

Binomials

A binomial is a polynomial with two terms. In our example, once the difference of squares is factored, the expression results in two binomials:

$(2a+3 + 5b)$
$(2a+3 - 5b)$

Binomials are written in the form $(A + B)$ or $(A - B)$.

Factoring into binomials is a common technique in algebra, which helps simplify expressions and solve equations more easily.

When dealing with binomials, it鈥檚 important to understand operations such as addition, subtraction, and multiplication of these expressions. By factoring, you can simplify complex algebra problems and make calculations easier.

Polynomials

Polynomials are expressions made up of variables and coefficients, involving operations of addition, subtraction, and multiplication. Our original expression, $(2a+3)^2 - 25b^2$, is a polynomial.

Polynomials are categorized by the number of terms:

a polynomial with two terms is called a binomial,
three terms make a trinomial,
and more than three terms are simply referred to as polynomials.

In the case of difference of squares, we typically start with a polynomial involving squared terms and reduce it via factoring.

Understanding how to work with polynomials is fundamental in algebra. Mastery of this concept allows you to tackle a range of mathematical problems, whether simplifying expressions or solving equations.

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Factor. $$(2 a+3)^{2}-25 b^{2}$$

Short Answer

Step by step solution

Identify the Squares

Apply the Difference of Squares Formula

Simplify the Expression

Key Concepts

Difference of Squares

Binomials

Polynomials

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