Problem 31 - Algebra A Combined Function

Question

Multiply. $$ (5 x+9)^{2} $$

Step-by-Step Solution

Answer

\((5x + 9)^2 = 25x^2 + 90x + 81\).

Step 1: Understand the Formula

To expand a squared binomial like \((a + b)^2\), you should use the formula: \((a + b)^2 = a^2 + 2ab + b^2\). This formula will help in expanding the expression by distributing and simplifying.

Step 2: Identify the Values of a and b

In the expression \((5x + 9)^2\), we identify \(a = 5x\) and \(b = 9\). We will use these values to plug into the expansion formula.

Step 3: Calculate a²

Calculate \(a^2\) by squaring the term \(5x\). So, \((5x)^2 = 25x^2\). This represents the first part of our expanded expression.

Step 4: Calculate 2ab

Next, calculate \(2ab\) where \(a = 5x\) and \(b = 9\). So, \(2ab = 2 \times 5x \times 9 = 90x\).

Step 5: Calculate b²

Find \(b^2\) by squaring the term \(9\). Therefore, \(9^2 = 81\). This is the third part of the expansion.

Step 6: Write the Expanded Expression

Combine all the parts from the previous steps to write out the full expanded expression: \((5x + 9)^2 = 25x^2 + 90x + 81\).

Key Concepts

Squared Binomials

When working with squared binomials, you're dealing with expressions that are in the form of \((a + b)^2\). Squaring a binomial is different from merely multiplying out each term separately. This specific operation is simplified by using a standard formula:

• \((a + b)^2 = a^2 + 2ab + b^2\)

Using this formula, you can quickly expand the expression without extensive manual multiplication.
Consider the example \((5x + 9)^2\). Here, the values of \(a\) and \(b\) are precisely identified as \(5x\) and \(9\) respectively, allowing for seamless application of the formula. This approach not only saves time but ensures the accuracy of the expansion process.

Binomial Expansion

Binomial expansion refers to the process of expressing binomial expressions raised to a power as a polynomial. The essential concept relies on distributing each component systematically according to established rules.
To solve \((5x + 9)^2\), we plug \(a = 5x\) and \(b = 9\) into the expansion formula to find each term individually:

• \(a^2 = (5x)^2 = 25x^2\)
• \(2ab = 2 \times 5x \times 9 = 90x\)
• \(b^2 = 9^2 = 81\)

These calculated terms are then combined to represent the binomial's expanded form: \(25x^2 + 90x + 81\). This method provides an efficient way to tackle squared binomials and facilitate polynomial calculation by focusing on one term at a time.

Polynomial Expansion

Polynomial expansion involves writing a product of binomials as a sum of terms with individual powers. It's a fundamental skill in algebra that is used for simplifying expressions and solving equations.
In expanding the expression \((5x + 9)^2\), each term such as \(5x\) and \(9\) is multiplied and combined to achieve the expanded polynomial form:

• \(25x^2\) resulting from \((5x)^2\)
• \(90x\) derived from \(2 \times 5x \times 9\)
• \(81\) calculated from \(9^2\)

Combining these gives the complete polynomial \(25x^2 + 90x + 81\), making polynomial expansion a powerful tool for converting binomial expressions into more manageable forms. Understanding this process is crucial when working with more complex algebraic expressions or preparing for calculus.