Problem 19 - Elementary Algebra

Question

For the following problems, find the products. $$ (b+15)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression \((b + 15)^2\) is \(b^2 + 30b + 225\).

Step 1: Identify the binomial terms

In this problem, the binomial expression is \((b + 15)^2\). The terms 'a' and 'b' in this case are 'b' and '15' respectively.

Step 2: Apply the binomial square formula

Using the formula mentioned above, we will expand \((b + 15)^2\): $$(b+15)^2 = b^2 + 2(b)(15) + 15^2$$

Step 3: Calculate the products

Now, we will perform the multiplications: $$b^2 + 30b + 225$$

Step 4: Write the final product

The final product after simplifying the given expression is: $$(b+15)^2 = b^2 + 30b + 225$$

Key Concepts

Binomial Expansion

Binomial expansion is a method used in algebra that allows us to expand expressions which are raised to a power. Specifically, binomial expansion is applied to expressions of the form \((a + b)^n\), where "a" and "b" are any terms, and "n" is a positive integer. The binomial expansion results in a polynomial that includes terms involving products of powers of "a" and "b".To expand this, we often use the Binomial Theorem, though for simpler cases like squares, there are shortcut formulas such as the binomial square formula:

• \((a + b)^2 = a^2 + 2ab + b^2\)

This formula makes it quick to expand the expression by calculating the square of each term and twice the product of the two terms.Knowing how to apply this technique is immensely useful when working with quadratic expressions or solving equations involving polynomials.

Polynomials

In mathematics, polynomials are expressions consisting of variables, coefficients, and exponents. They can take many forms, but basically, they are sums of terms of the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where each \(a_i\) is a coefficient and \(x\) is a variable raised to an exponent "n".Polynomials are classified by their degree, which is the highest power of the variable present in the polynomial:

• A polynomial of degree 2 is called a quadratic,
• Degree 3 is a cubic, and so on.

Polynomials are the building blocks of algebra and are used in functions, models, and various types of equations. Understanding their structure and manipulation allows you to tackle more complex mathematical concepts effectively.

Exponentiation

Exponentiation is the mathematical operation involving numbers called "base" and "exponent." It is expressed as \(a^n\) where "a" is the base and "n" is the exponent, indicating how many times the base multiplies by itself.This operation is central to many algebraic processes. It simplifies the writing of repeated multiplication. For instance, \(b^2 = b \times b\). It also introduces properties that assist in calculations, such as:

• \(a^m \times a^n = a^{m+n}\) - when multiplying same bases, add exponents,
• \(\frac{a^m}{a^n} = a^{m-n}\) - when dividing same bases, subtract exponents,
• \((a^m)^n = a^{m \times n}\) - multiplying exponents when raising a power to a power.

Exponentiation is widely used, from calculating compound interest in finance to determining bacterial growth in biology, offering a compact way to express complex multiplicative relationships.