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Chapter 5: Problem 58

Factor completely. \(4 y^{5}+12 y^{4}-40 y^{3}\)

Short Answer

Expert verified
Complete factorization: \( 4 y^{3}(y + 5)(y - 2) \)

Step by step solution

01

Factor out the greatest common factor (GCF)

Identify and factor out the greatest common factor from all terms in the expression. The GCF of the terms in the expression is 4y鲁. o \[ 4 y^{5} + 12 y^{4} - 40 y^{3} = 4 y^{3}(y^{2} + 3 y - 10) \]
02

Factor the quadratic expression

Now, factor the quadratic expression inside the parentheses: \( y^{2} + 3y - 10 \). Look for two numbers that multiply to -10 (constant term) and add up to 3 (coefficient of the middle term). These numbers are 5 and -2. Therefore, the expression factors into: \( (y + 5)(y - 2) \)
03

Combine the factors

Combine the factored quadratic expression with the GCF factored out in Step 1. The complete factorization of the expression is: \[ 4 y^{5} + 12 y^{4} - 40 y^{3} = 4 y^{3}(y + 5)(y - 2) \]

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Key Concepts

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

greatest common factor
The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. When factoring polynomials, we start by finding the GCF of all the terms in the polynomial. This simplifies the polynomial, making it easier to factor further.
factor quadratic expressions
Next, we factor the quadratic expression inside the parentheses. The quadratic expression is in the standard form: ax^2 + bx + c. We need to find two numbers that multiply to the constant term (c) and add up to the coefficient of the middle term (b).
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations. Understanding how to manipulate and factor these expressions is fundamental in algebra.

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Most popular questions from this chapter

Factor completely. \(m^{3} n-10 m^{2} n^{2}+24 m n^{3}\)

Factor polynomial. \((2 p+q) r^{2}-12(2 p+q) r+27(2 p+q)\)

Solve each equation. $$ 2 t+10=0 $$

Factor polynomial. \(m^{3} n-2 m^{2} n^{2}-3 m n^{3}\)

Use this method to factor each trinomial. $$ -2 a^{2}-5 a b-2 b^{2} $$
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