/*! 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 37 In the following exercises, fact... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 37

In the following exercises, factor by grouping. $$ a b+5 a+3 b+15 $$

Short Answer

Expert verified
(b + 5)(a + 3)

Step by step solution

01

Group the terms

Group the terms into two pairs to make factoring easier. Rewrite the expression as: a b + 5 a + 3 b + 15 can be grouped into a b + 5 a and 3 b + 15.
02

Factor out the common factors from each group

In the first group, factor out the common factor 'a': a(b + 5) In the second group, factor out the common factor '3': 3(b + 15/3) Since 15/3 = 5, the expression becomes: a(b + 5) + 3(b + 5)
03

Factor out the common binomial

Notice that (b + 5) is a common binomial factor in both terms. Factor (b + 5) out of the expression: a(b + 5) + 3(b + 5) becomes (b + 5)(a + 3)

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.

algebraic expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. For example, in the exercise provided, the expression is: \[ ab + 5a + 3b + 15 \].
Here, 'a' and 'b' are variables while '5', '3', and '15' are constants.
We can combine these elements through operations like addition and multiplication.
Understanding how to group terms and identify common factors within these expressions is crucial for simplifying and solving them.
common factors
Common factors are numbers or terms that are shared by two or more expressions.
To factor by grouping, we first rewrite the given expression: \[ ab + 5a + 3b + 15 \]
We can group the terms into pairs: \[ (ab + 5a) + (3b + 15) \]
Next, we find the common factors within each group:
For the first group \[ (ab + 5a) \], the common factor is 'a', so we factor it out: \[ a(b + 5) \]
For the second group \[ (3b + 15) \], the common factor is '3', so we factor it out: \[ 3(b + 5) \].
Now, we see that the binomial \[ (b + 5) \] is common to both groups.
binomial factoring
Binomial factoring involves factoring out a common binomial from an algebraic expression.
In our exercise, after grouping and extracting common factors, we are left with: \[ a(b + 5) + 3(b + 5) \].
Notice that \[ (b + 5) \] is a common binomial factor.
We factor out \[ (b + 5) \] to combine the terms: \[ (b + 5)(a + 3) \].
This final expression is the product of two binomials and represents the fully factored form of the original expression.
Understanding how to identify and factor out common binomials is key in simplifying complex algebraic expressions.

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

Factor completely using the sums and differences of cubes pattern, if possible. $$ 27 y^{3}+8 z^{3} $$

Factor completely using the difference of squares pattern, if possible. $$ 98 r^{3}-72 r $$

Explain what it mean to factor a polynomial completely.

Solve. Gianna is going to throw a ball from the top floor of her middle school. When she throws the ball from 48 feet above the ground, the function \(h(t)=-16 t^{2}+32 t+48\) models the height, \(h\), of the ball above the ground as a function of time, \(t\). Find: (a) the zeros of this function which tells us when the ball will hit the ground. (b) the time(s) the ball will be 48 feet above the ground. (c) the height the ball will be at \(t=1\) seconds which is when the ball will be at its highest point.

Solve. A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.