/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Factor each trinomial. \(a^{2}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 25

Factor each trinomial. \(a^{2}-2 a b-35 b^{2}\)

Short Answer

Expert verified
The factored form is (a - 7b)(a + 5b).

Step by step solution

01

- Identify coefficients

The given trinomial is: a^{2} - 2ab - 35b^{2}Identify the coefficients: a^{2} (Coefficient of a^2): 1 -2ab (Coefficient of ab): -2 -35b^{2} (Coefficient of b^2): -35
02

- Find two numbers that multiply to the constant term

Find two numbers that multiply to the constant term -35 and add up to the middle coefficient -2.The two numbers are: -7 and 5 since -7 * 5 = -35 and -7 + 5 = -2
03

- Rewrite the middle term

Rewrite the middle term -2ab using the two numbers found:a^{2} - 7ab + 5ab - 35b^{2}
04

- Group and factor by grouping

Group the terms into two pairs and factor each group:(a^{2} - 7ab) + (5ab - 35b^{2}) Factor out the greatest common factor (GCF) from each group:a(a - 7b) + 5b(a - 7b)
05

- Factor out the common binomial factor

Notice that (a - 7b) is a common factor:(a - 7b)(a + 5b)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

headline of the respective core concept
Factoring trinomials can seem daunting at first, but once you understand the steps, it becomes much simpler. The key here is recognizing the structure of a trinomial, typically written as \[ax^2 + bx + c\]. To factor this, you'll follow a systematic method to break it down into the product of two binomials. Let's break down the process.
headline of the respective core concept
The term 'coefficients' in a polynomial are the numbers in front of the variables. For example, in the trinomial \[a^{2}-2ab-35b^{2}\], each term has a coefficient:
  • For \[a^2\], the coefficient is 1.
  • For \[-2ab\], the coefficient is -2.
  • For \[-35b^2\], the coefficient is -35.
Identifying these coefficients is crucial as they guide you in the factoring process. By knowing these, you can find patterns and determine which pairs of numbers will help you factor the trinomial effectively.
headline of the respective core concept
The greatest common factor (GCF) is the largest factor that divides two or more numbers. When factoring trinomials, finding the GCF can simplify the process. In our example, after rewriting \[-2ab\] and factoring by grouping, we saw:
  • \[a(a - 7b)\]
  • \[5b(a - 7b)\]
This reveals that \[a - 7b\] is a common binomial factor. Factoring out this common factor simplifies the trinomial into the product of binomials, leading to our final factorized expression: \[(a - 7b)(a + 5b)\]. Understanding how to factor out the GCF helps in simplifying and ultimately solving trinomial equations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve each equation. $$ -x^{2}-8 x=16 $$

Factor each trinomial. \(12 p^{6}-32 p^{3} r+5 r^{2}\)

Solve each equation. (Hint: In Exercises \(51-54,\) use the substitution of variable method.) $$ 2(x-1)^{2}-7(x-1)-15=0 $$

Solve each equation. (Hint: In Exercises \(51-54,\) use the substitution of variable method.) $$ 5(3 x-1)^{2}+3=-16(3 x-1) $$

A square mirror has sides measuring \(2 \mathrm{ft}\) less than the sides of a square painting. If the difference between their areas is \(32 \mathrm{ft}^{2},\) find the lengths of the sides of the mirror and the painting.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.