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📘intermediate algebra review

Perfect Square Binomial

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

A perfect square binomial is a special type of polynomial expression that can be factored into the square of a single term. It consists of two terms, one of which is the square of a variable or number, and the other term is twice the product of the square root of the first term and a second term.

5 Must Know Facts For Your Next Test

  1. The general form of a perfect square binomial is $a^2 + 2ab$ or $a^2 - 2ab$, where $a$ and $b$ are variables or numbers.
  2. Perfect square binomials can be easily factored by identifying the square of the first term and twice the product of the square root of the first term and the second term.
  3. Recognizing perfect square binomials is important in the context of multiplying polynomials, as they can be factored to simplify the multiplication process.
  4. Perfect square binomials are commonly used in algebraic expressions and equations, and understanding their properties can aid in solving various mathematical problems.
  5. The ability to identify and manipulate perfect square binomials is a crucial skill in intermediate algebra, as it allows for efficient factorization and simplification of polynomial expressions.

Review Questions

  • Explain the general form of a perfect square binomial and how it can be factored.
    • The general form of a perfect square binomial is $a^2 + 2ab$ or $a^2 - 2ab$, where $a$ and $b$ are variables or numbers. To factor a perfect square binomial, you identify the square of the first term ($a^2$) and twice the product of the square root of the first term and the second term ($2ab$). This allows you to express the binomial as the square of a single term, such as $(a + b)^2$ or $(a - b)^2$.
  • Describe the significance of perfect square binomials in the context of multiplying polynomials.
    • Perfect square binomials are important in the context of multiplying polynomials because they can be factored to simplify the multiplication process. By recognizing a perfect square binomial, you can quickly factor it and then multiply the factored terms, rather than multiplying the entire polynomial expression. This streamlines the multiplication of polynomials and can lead to more efficient algebraic manipulations.
  • Analyze how the properties of perfect square binomials can be applied to solve various mathematical problems.
    • The properties of perfect square binomials can be applied to solve a wide range of mathematical problems, such as simplifying algebraic expressions, solving equations, and working with geometric formulas. For example, recognizing a perfect square binomial in an equation can allow you to factor it and solve for the unknown variable more easily. Additionally, the square root property of perfect square binomials can be used to simplify expressions involving radicals. Understanding the characteristics of perfect square binomials is a valuable skill that can be applied across different areas of intermediate algebra.