Explanation Procedure used: Factoring general trinomials by grouping : In a trinomial of the form a x 2 + b x + c , a ≠ 1 , split up the x -term into two terms using the method described below. “Step 1: Find the product a c . Step 2: Find two numbers whose product is a c and whose sums is b . Step 3: Replace the middle term bx with two x -terms having these numbers as coefficients. That is, a x 2 + b x + c = a x 2 + m x + n x + c . Step 4: Complete the factorization by grouping”. Calculation: Factor the given expression a x 3 + 4 a 2 x 2 − 12 a 3 x as follows. Rewrite the expression as, a x 3 + 4 a 2 x 2 − 12 a 3 x = a x ( x 2 + 4 a x − 12 a 2 ) (1) Consider the expression x 2 + 4 a x − 12 a 2 . It is observed that, the coefficients are a = 1 , b = 4 a and c = − 12 a 2 . Step 1: Find the value of a c . Obtain the value of a c as, a c = 1 ( − 12 a 2 ) = − 12 a 2 Thus, the value of a c is − 12 a 2 . Step 2: Find the two number, e and f, whose product is a c = − 12 a 2 and whose sum is b = 7 . Obtain the list of integer factors of − 12 a 2 such that product is − 12 a 2 and sum is 4 a as shown below in Table 1. Integers whose product is − 12 a 2 a , − 12 a − a , 12 a 6 a , − 2 a − 6 a , 2 a Sum –11 a 11 a 4 a –4 a Table 1 From Table 1, it is observed that the numbers whose product is − 12 a 2 and sum is 4 a are − 2 a and 6 a . Let the variables e = − 2 a and f = 6 a

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 6.2, Problem 51E

To determine

To factor: The expression ax3+4a2x2−12a3x completely.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.2, Problem 50E

Chapter 6.2, Problem 52E

Students have asked these similar questions

7. [10 marks] Let G = (V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a cycle in G on which x, y, and z all lie. (a) First prove that there are two internally disjoint xy-paths Po and P₁. (b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that there are three paths Qo, Q1, and Q2 such that: ⚫each Qi starts at z; • each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are distinct; the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex 2) and are disjoint from the paths Po and P₁ (except at the end vertices wo, W1, and w₂). (c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and z all lie. (To do this, notice that two of the w; must be on the same Pj.)

6. [10 marks] Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of T. (a) How many vertices does BL(T) have? (b) How many edges does BL(T) have? Prove that your answers are correct.

4. [10 marks] Find both a matching of maximum size and a vertex cover of minimum size in the following bipartite graph. Prove that your answer is correct. ย ພ

Chapter 6 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

Ch. 6.1 - Prob. 1PE Ch. 6.1 - Prob. 2PE Ch. 6.1 - Prob. 3PE Ch. 6.1 - Prob. 4PE Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

To factor: The expression a x 3 + 4 a 2 x 2 − 12 a 3 x completely.

Videos

To factor: The expression ax3+4a2x2−12a3x completely.

Want to see the full answer?

Chapter 6 Solutions