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Chapter 5: Problem 56

Write an equivalent expression by factoring. $$2 y^{5}+15-6 y^{4}-5 y$$

Short Answer

Expert verified
\( (2y^4 - 5)(y - 3) \)

Step by step solution

01

Rearrange terms

Start by rearranging the terms of the polynomial in descending order of the exponents: \[2y^5 - 6y^4 - 5y + 15\]
02

Factor by grouping

Group the terms in pairs to factor separately: \[(2y^5 - 6y^4) + (-5y + 15)\]
03

Factor out the greatest common factor (GCF) from each group

Factor out the GCF from each pair: \[2y^4(y - 3) - 5(y - 3)\]
04

Factor out the common binomial factor

Notice that \(y - 3\) is a common factor in both terms. Factor \(y - 3\) out: \[(2y^4 - 5)(y - 3)\]
05

Write the final factored form

The expression is now factored completely as: \[(2y^4 - 5)(y - 3)\]

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Key Concepts

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

polynomial
A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomials can take many forms, but they always follow these rules:
  • Only non-negative integer powers are allowed.
  • Coefficients can be any real number.
  • Examples: 3x虏 + 2x - 5, 4y^3 - y + 7.
In our example, the polynomial expression we need to factor is: 2y^5 + 15 - 6y^4 - 5y.
The first step in the process involves writing the polynomial in standard form, which means arranging the terms in descending order of their exponents. If we do this, we get: 2y^5 - 6y^4 - 5y + 15.
factoring by grouping
Factoring by grouping is a method used when a polynomial has four or more terms. It involves grouping terms to make the polynomial easier to factor. Let's see how this works using our polynomial: 2y^5 - 6y^4 - 5y + 15.
  • First, divide the polynomial into groups: (2y^5 - 6y^4) and (-5y + 15).
  • Notice that within each group, we can factor out a common factor. For the first group, 2y^4 is the GCF, and for the second group, 5 is the GCF:
    • (2y^4 * y - 2y^4 * 3) - 5(y - 3).
  • Now, we write the factored form of each group: 2y^4(y - 3) - 5(y - 3).
    • Next, we see that y - 3 is a common factor in both groups, and we factor y - 3 out from both terms: (2y^4 - 5)(y - 3).
Therefore, factoring by grouping gives us the final factored form of: (2y^4 - 5)(y - 3).
greatest common factor
The greatest common factor (GCF) is the largest factor that divides two or more numbers. To factor a polynomial by grouping, we need to find the GCF of each group of terms.
Here鈥檚 how to find it:
  • Identify the common factors in each group. For instance, in the first group (2y^5 - 6y^4), the GCF is 2y^4.
    • In the second group (-5y + 15), the GCF is 5.
  • This means we can write: 2y^4(y - 3) - 5(y - 3).
The GCF plays a crucial role because it simplifies the polynomial, allowing us to factor out the common binomial factor from the groups, which leads ultimately to the completely factored polynomial: (2y^4 - 5)(y - 3).
Always remember, finding the GCF is essential to the factoring process as it makes the polynomial expressions simpler and more manageable.

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