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Chapter 6: Problem 168

Factor completely using the perfect square trinomials pattern. $$ 25 r^{2}+60 r s+36 s^{2} $$

Short Answer

Expert verified
\((5r + 6s)^2\)

Step by step solution

01

Identify the perfect square terms

In the trinomial expression, identify the terms that are perfect squares. Here, 25 and 36 are perfect squares since: \(25r^2 = (5r)^2\) and \(36s^2 = (6s)^2\).
02

Verify the middle term

Check if the middle term, 60rs, fits the pattern of \(2ab\) where \(a = 5r\) and \(b = 6s\). Calculate: \(2 \times 5r \times 6s = 60rs\), which matches the middle term.
03

Write the trinomial as a binomial squared

Since the trinomial fits the perfect square pattern, write it as a squared binomial: \((5r + 6s)^2\).

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Key Concepts

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

Factoring
Factoring in algebra involves breaking down a complex expression into simpler parts that, when multiplied together, give the original expression. It's like reverse engineering. Given an equation or polynomial, factoring makes it easier to solve.

For example, for the expression 25r虏 + 60rs + 36s虏, the goal is to find expressions within it that multiply together to result in the same equation. Factoring helps in simplifying and solving equations, especially quadratic equations.

There are different factoring techniques, such as:
  • Factoring out the greatest common factor (GCF)
  • Factoring by grouping
  • Factoring trinomials
  • Special formulas like perfect square trinomials and difference of squares
Understanding and mastering these methods is crucial for solving complex algebraic problems efficiently.
Algebra
Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. The symbols represent variables that can stand for unknown numbers, allowing us to solve equations and understand relationships.

In algebra, terms and operations work together to form expressions that can be simplified or solved. For example, in the expression 25r虏 + 60rs + 36s虏, each element like 25r虏, 60rs, and 36s虏 is a term. We use algebraic techniques to manipulate these terms.

Key concepts in algebra include:
  • Variables and constants
  • Expressions and equations
  • Factoring
  • Expanding and simplifying expressions
  • Solving linear and quadratic equations
Understanding algebra is fundamental because it forms the basis for higher-level math and various real-world problem-solving scenarios.
Perfect Square Trinomials
A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. Recognizing this pattern can simplify the factoring process.

For the trinomial 25r虏 + 60rs + 36s虏, we need to identify if it fits the pattern of a perfect square trinomial. Follow these steps:
  • Identify the square terms: 25r虏 and 36s虏 are perfect squares because \(25r^2 = (5r)^2\) and \(36s^2 = (6s)^2\).
  • Check the middle term: The middle term, 60rs, should match the pattern 2ab where \(a = 5r\) and \(b = 6s\). Ensure \(2 \times 5r \times 6s = 60rs\) fits the given middle term.
  • Write as a squared binomial: Since the criteria are met, the trinomial can be written as \( (5r + 6s)^2 \).
Recognizing and practicing the factoring of perfect square trinomials helps in solving quadratic equations more efficiently. It's a fundamental algebraic skill useful in both simple and complex mathematical problems.

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Most popular questions from this chapter

In the following exercises, factor using the 'ac' method. $$ 6 p^{2}+p-22 $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x y+c y^{2} .\) $$ r^{2}+3 r s-28 s^{2} $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x y+c y^{2} .\) $$ m^{2}-64 m n-65 n^{2} $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ y^{2}-18 y+45 $$

In the following exercises, factor completely using trial and error. $$ 11 n^{3}-55 n^{2}+44 n $$
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