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Chapter 4: Problem 61

Find each product. $$ (y-5)^{2} $$

Short Answer

Expert verified
The product is \(y^2 - 10y + 25\).

Step by step solution

01

Write down the expression

Write down the given expression: \((y - 5)^2\). This is a binomial squared.
02

Apply Binomial Theorem

Use the Binomial Theorem for squaring a binomial, which states: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = y\) and \(b = 5\).
03

Square the first term

Square the first term of the binomial: \(y^2\).
04

Multiply and double the terms

Multiply the two terms \(y\) and \(5\) and then double the result: \(-2 \times y \times 5 = -10y\).
05

Square the second term

Square the second term of the binomial: \(5^2 = 25\).
06

Combine all parts

Combine all parts from Steps 3, 4, and 5 to get: \(y^2 - 10y + 25\).

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Key Concepts

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

Squaring binomials
Squaring binomials is a common operation in algebra that you will often encounter. A binomial is a polynomial with exactly two terms. When we square a binomial, we are raising it to the power of 2. For example, \((y - 5)^2\). To square a binomial, we use the formula: \[ (a - b)^2 = a^2 - 2ab + b^2 \]. This means, for \((y - 5)^2\):
  • First, square the first term: \((y)^2 = y^2\).
  • Next, multiply the two terms and double the result: \(-2 \times y \times 5 = -10y\).
  • Finally, square the second term: \(5^2 = 25\).
Combining all these parts, we obtain \(y^2 - 10y + 25\). This process makes squaring binomials quick and straightforward once you memorize the formula.
Polynomial expansion
Polynomial expansion involves expressing a polynomial as the sum of its individual terms. When dealing with binomials, the Binomial Theorem is a useful tool for expansion. For example, in \(y - 5)^2\), we applied the Binomial Theorem to expand it: \[ (a - b)^2 = a^2 - 2ab + b^2 \]. This process breaks down a complex expression into simpler, easier-to-handle terms. Here are the steps:
  • Identify each term in the binomial, where \(a = y\) and \(b = 5\).
  • Expand the squared binomial using the formula: \(a^2 - 2ab + b^2\).
The Binomial Theorem can be extended for higher powers, but the fundamental idea remains the same: expanding the polynomial into the sum of individual terms.
Algebraic expressions
Algebraic expressions are mathematical phrases that can contain variables, numbers, and operations. They are the building blocks of algebra and are used to represent real-world situations. In the exercise above, \((y - 5)^2\) is an algebraic expression containing a binomial.
  • Breaking down the expression helps simplify calculations and solve problems.
  • The expanded form \(y^2 - 10y + 25\) is also an algebraic expression.
  • These expressions help in finding values, solving equations, and modeling scenarios in math and science.
Understanding how to manipulate these expressions鈥攕uch as through squaring a binomial or expanding a polynomial鈥攅nables you to handle a wide range of algebraic problems more effectively.

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