Problem 59 - Algebra 1: Concepts and Skills

Question

Find the product of \((3 x+5)^{2}\) $$F. 3 x^{2}+15 x+5$$ $$G. 9 x^{2}+25$$ $$H.3 x^{2}+25$$ $$J. 9 x^{2}+30 x+25$$

Step-by-Step Solution

Answer

So, the product of \((3 x+5)^{2}\) is \(9 x^{2}+30 x+25\), which corresponds to option J.

Step 1: Write down the formula for the square of a binomial

The square of a binomial \((a+b)^2\) is given by the formula \(a^2 + 2ab + b^2\).

Step 2: Substitute the values of \(a\) and \(b\) from the binomial \(3x+5\) into the formula

Here, \(a = 3x\) and \(b = 5\). Substitute these into the above formula to get \((3x)^2 + 2*(3x)*5 + (5)^2\).

Step 3: Simplify the expression

Simplify the above expression to get \(9x^2 + 30x + 25\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are the backbone of algebra and comprise variables, constants, and arithmetic operations. For instance, ewline ewline ewline ewline in the product ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

Applying the Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that provides a quick way to expand polynomials raised to a power. This theorem tells us that squaring a binomial like ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

Executing Polynomial Multiplication

Multiplying polynomials, like the square of a binomial we're considering, is a fundamental algebraic technique. Using the distributive property—a cornerstone of polynomial operations—the product is found by multiplying each term in one binomial by every term in the other. ewline ewline For the expression ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline