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Chapter 5: Problem 56

Factor each trinomial. \(4 p^{3}+24 p^{2}-64 p\)

Short Answer

Expert verified
4p(p + 8)(p - 2)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Look for the greatest common factor in the terms of the quadratic. All terms in the trinomial have a common factor of 4p. So, factor out the GCF first: 4p(p^2 + 6p - 16)
02

Factor the Remaining Trinomial

Now focus on factoring the remaining trinomial: p^2 + 6p - 16. We need to find two numbers that multiply to -16 and add up to 6. After identifying the numbers 8 and -2, we can factor the trinomial: (p + 8)(p - 2)
03

Write the Fully Factored Form

Combine the GCF with the factored trinomial to get the final answer: 4p(p + 8)(p - 2)

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Key Concepts

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

Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is a fundamental concept in factoring trinomials.
The GCF is the largest factor that divides all terms in an expression without leaving a remainder.
For example, in the trinomial provided (4 p^{3}+24 p^{2}-64 p), we first identify the GCF of all terms.
Here, 4p is the greatest factor common to all the terms.
This means we can factor out 4p from each term:
4p(p^2 + 6p - 16).

Understanding how to find the GCF can simplify the problem significantly, making it easier to work with the remaining trinomial.
Factoring
Factoring is the process of breaking down an expression into products of simpler factors, which is essential in solving many algebraic problems.
Once the GCF is factored out, we are left with a simpler trinomial to factor: p^2 + 6p - 16.
To factor this, we need to identify two numbers that multiply to -16 (the constant term) and add up to 6 (the coefficient of the middle term).
After considering the factors, we find that the numbers 8 and -2 fulfill these requirements:
(p + 8)(p - 2).

Therefore, we have successfully factored the trinomial. By understanding each step, you can apply this method to different problems.
Quadratic Equations
Working with quadratic equations often involves factoring, especially when solving for roots.
A quadratic equation is typically in the form ax^2 + bx + c = 0.
In this example, after factoring out the GCF, we were left with the quadratic equation in standard form: p^2 + 6p - 16 = 0.
Factoring this quadratic into (p + 8)(p - 2), we can easily solve for the roots by setting each factor equal to zero:
p + 8 = 0 or p - 2 = 0.
This gives us the solutions: p = -8 and p = 2.

Understanding quadratic equations and their factoring is crucial for problem-solving in algebra and higher-level math.

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