Problem 54 Factor each trinomial. (Hint: Fa... [FREE SOLUTION]

Chapter 6: Problem 54

Factor each trinomial. (Hint: Factor out the GCF first.) $$ m^{2}(m-p)^{2}-m p(m-p)^{2}-12 p^{2}(m-p)^{2} $$

Short Answer

Expert verified

$ (m-p)^{2} (m + 3p)(m - 4p) $

Step by step solution

- Factor out the common term

Look at the given trinomial: $ m^{2}(m-p)^{2}-m p(m-p)^{2}-12 p^{2}(m-p)^{2} $. Notice that $(m-p)^{2}$ is common in all three terms. Factor $(m-p)^{2}$ out: \[ (m-p)^{2} [m^{2} - mp - 12p^{2}] \]

- Factor the quadratic expression

Next, we need to factor the quadratic expression $m^{2} - mp - 12p^{2}$. We look for two numbers that multiply to $-12p^{2}$ and add to $-p$. These numbers are $-4p$ and $3p$. The quadratic expression can be factored as: \[ m^{2} - 4mp + 3mp - 12p^{2} \].

- Group the terms and factor

Now group the terms and factor by grouping: \[ m(m - 4p) + 3p(m - 4p) \]. Notice that $(m - 4p)$ is common in both groups. So, factor $(m - 4p)$ out: \[ (m + 3p)(m - 4p) \].

- Combine all factored parts

Combine the factored quadratic with the previously factored common term: \[ (m-p)^{2} (m + 3p)(m - 4p) \]. This is the fully factored form of the trinomial.

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.

greatest common factor

When factoring trinomials, finding the greatest common factor (GCF) is a crucial first step. The GCF is the largest factor that divides all terms in the polynomial. In our example, the trinomial is:

$$m^{2}(m-p)^{2}-m p(m-p)^{2}-12 p^{2}(m-p)^{2}$$.

We can see that $(m-p)^{2}$ is present in all terms. Therefore, $(m-p)^{2}$ is the GCF. By factoring out this common term, we simplify the trinomial significantly:

$(m-p)^{2} [m^{2} - mp - 12p^{2}]$.
This makes the next steps easier and more manageable.

factoring quadratics

Factoring quadratic expressions is a key step in factoring trinomials. Quadratics take the form of $ax^2 + bx + c$. In our example, after factoring out $(m-p)^{2}$, we are left with the quadratic expression $m^{2} - mp - 12p^{2}$. To factor this, we need to find two numbers that multiply to the constant term ($-12p^2$), and add to the coefficient of the middle term (-p). These numbers are$-4p$ and $3p$, because$-4p * 3p = -12p^2$ and $-4p + 3p = -p$.

Therefore, we can rewrite the quadratic as: $m^{2} - 4mp + 3mp - 12p^{2}$.

factoring by grouping

After rewriting the quadratic expression, we employ a technique called factoring by grouping. This means to group terms in such a way that each group has a common factor. In our case:

m(m - 4p) + 3p(m - 4p).

Notice that ($m - 4p$) appears in both groups. This allows us to factor out ($m - 4p$) from the expression:

(m - 4p)(m + 3p). Grouping makes it easier to factor more complex expressions by breaking them into simpler parts.

polynomial factoring

Finally, we combine all the factored parts to get the fully factored form of the original trinomial. From the previous steps, we have: $(m-p)^{2} [m^{2} - mp - 12p^{2}]$, and we factored the quadratic to get $(m + 3p)(m - 4p)$.

So, combining everything: $(m-p)^{2} (m + 3p)(m - 4p)$.

This is the completely factored form of the given trinomial, having factored out the GCF and broken down the quadratic expression through grouping and polynomial factoring.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Factor each trinomial. (Hint: Factor out the GCF first.) $$ m^{2}(m-p)^{2}-m p(m-p)^{2}-12 p^{2}(m-p)^{2} $$

Short Answer

Step by step solution

- Factor out the common term

- Factor the quadratic expression

- Group the terms and factor

- Combine all factored parts

Key Concepts

greatest common factor

factoring quadratics

factoring by grouping

polynomial factoring

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Pure Maths

Decision Maths

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.