Problem 89 In the following exercises, fact... [FREE SOLUTION]

Chapter 6: Problem 89

In the following exercises, factor completely using trial and error. $$ p^{3}-8 p^{2}-20 p $$

Short Answer

Expert verified

p(p - 10)(p + 2)

Step by step solution

Identify common factors

First, look for any common factors in the given polynomial. The polynomial is p^{3} - 8p^{2} - 20p. Each term contains a factor of p.

Factor out the greatest common factor (GCF)

Factor out p from each term. This gives: p(p^{2} - 8p - 20).

Factor the quadratic expression

Now, factor the quadratic expression within the parenthesis: p^{2} - 8p - 20. To do this, look for two numbers that multiply to -20 (the constant term) and add to -8 (the coefficient of the linear term). These numbers are -10 and 2.

Write the factored form

Express the quadratic as a product of binomials: (p - 10)(p + 2). Then, multiply by the factored-out p from Step 2. This gives the completely factored form: p(p - 10)(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

Factoring polynomials often starts by identifying the greatest common factor (GCF). The GCF is the largest factor that divides each term of the polynomial.
For example, in the problem p^{3} - 8p^{2} - 20p, each term contains the factor 'p'. So, by factoring out 'p', we simplify the polynomial to p(p^{2} - 8p - 20).
This step reduces the complexity of the polynomial, making it easier to apply further factoring techniques.

quadratic expressions

Quadratic expressions are polynomials of degree 2, typically in the form ax^{2} + bx + c. Once the GCF is factored out, we often need to factor a quadratic expression.
In our example, this quadratic expression is p^{2} - 8p - 20. Factoring quadratics involves finding two numbers that multiply to the constant term (c) and add to the coefficient of the linear term (b).
Here, we need two numbers that multiply to -20 and add to -8. These numbers are -10 and 2. The quadratic can then be expressed as (p - 10)(p + 2), following the form (p + m)(p + n).

binomials

A binomial is an algebraic expression with two terms. These can often be found within the process of factoring polynomials. In our example, after factoring the quadratic expression, we get two binomials: (p - 10) and (p + 2).
Binomials are easier to manage and can often reveal further factoring opportunities or simplify expressions for solving equations. To complete our factoring, we incorporate the previously factored-out GCF, giving the final form: p(p - 10)(p + 2).
Understanding how to break down to these smaller parts is crucial for tackling more complex polynomial factoring.

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91影视

In the following exercises, factor completely using trial and error. $$ p^{3}-8 p^{2}-20 p $$

Short Answer

Step by step solution

Identify common factors

Factor out the greatest common factor (GCF)

Factor the quadratic expression

Write the factored form

Key Concepts

greatest common factor

quadratic expressions

binomials

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