Problem 50 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (9-5 x)^{2} $$

Step-by-Step Solution

Answer

The result of squaring the binomial \( (9 - 5x)^{2} \) is \( 25x^{2} - 90x + 81 \)

Step 1: Identification

Firstly, identify what is a and what is b in the binomial formula. In this case, a is equal to 9 and b is equal to 5x.

Step 2: Application of formula

Now, apply the formula: \( (a - b)^{2} = a^{2} - 2ab + b^{2} \). So, this becomes: \( (9 - 5x)^{2} = 9^{2} - 2*9*5x + (5x)^{2} \).

Step 3: Calculation

Next, calculate the terms: \( 81 - 90x + 25x^{2} \).

Step 4: Rearrangement

Rearrange the terms so that the term with the highest degree of x is first, giving a final answer of \( 25x^{2} - 90x + 81 \).

Key Concepts

Polynomial

A polynomial is an expression made up of variables (like \( x \)) and coefficients, involving operations such as addition, subtraction, and multiplication. A common example of a polynomial is \( 25x^2 - 90x + 81 \), where each term is a product of a number and a power of \( x \). Polynomials can have different degrees, depending on the highest power of \( x \) present. In this case, the polynomial is of degree 2 because the highest power of \( x \) is 2. Polynomials are fundamental in algebra as they are used to describe relationships and to solve equations. Learning how to manipulate polynomials through operations like addition, subtraction, and multiplication is key to understanding deeper mathematical concepts.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. The symbols often represent numbers in equations. The primary focus is to find the unknown values that make the equation true. In the exercise given, we use algebraic principles to expand the expression \((9-5x)^2\). We applied the formula \((a - b)^2 = a^2 - 2ab + b^2\), which is a standard algebraic identity. This involves understanding how to handle expressions and equations systematically. Algebra is crucial because it provides a clear and universal way to solve problems, such as the ones involving polynomials, through the use of formulas and logical progression.

Factoring

Factoring is a technique used to break down a polynomial into simpler parts that, when multiplied together, give the original polynomial. It is essentially the reverse of expanding a polynomial. Though not directly applied in the current exercise, understanding factoring helps us recognize how complex polynomials can be simplified or broken up into simpler binomials. For instance, reversing the expansion of \( 25x^2 - 90x + 81 \) could lead us back to something like \((5x - 9)^2\), although here simplification stays primarily in the context of expanding binomials. Factoring allows solutions to be checked and verified quickly, making it a crucial skill in algebra up to more advanced topics in mathematics.