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Chapter 10: Problem 465

Factor the greatest common factor from each polynomial. $$55 y-11 y^{4}$$

Short Answer

Expert verified
11y(5 - y^3)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Find the greatest common factor of the coefficients and the variables. Here, the coefficients are 55 and 11, and the terms are 55y and -11y^4. The GCF of 55 and 11 is 11. For the variables y and y^4, the GCF is y.
02

Factor out the GCF from the polynomial

Write the polynomial as a product of the GCF and another polynomial. This means rewriting 55y - 11y^4 as 11y(5 - y^3).
03

Verify the factorization

Ensure the factorization is correct by distributing the GCF back through the polynomial. If 11y(5 - y^3) = 55y - 11y^4, the factorization is correct.

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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.
For example, in the polynomial 55y - 11y^4, the coefficients are 55 and 11, and the terms are 55y and -11y^4.
To find the GCF of these coefficients, list the factors of each number:
  • Factors of 55: 1, 5, 11, 55
  • Factors of 11: 1, 11
The greatest common factor of 55 and 11 is 11 because it is the highest number that divides both.
For the variables, we have y and y^4. The GCF for these variables is y since it is the lowest power of y present in both terms.
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors.
Using the polynomial 55y - 11y^4, we factor out the GCF which we found to be 11y.
This means rewriting the polynomial as the product of 11y and another polynomial.
Thus, we write:
11y(5 - y^3)
To find this factorization, you divide each term in the polynomial by the GCF:
  • 55y 梅 11y = 5
  • -11y^4 梅 11y = -y^3
Putting it together, we get 11y(5 - y^3).
Remember that polynomial factorization simplifies expressions and can make further mathematical operations easier.
Distributive Property
The distributive property is a useful algebraic property that allows you to multiply a common factor across terms within parentheses.
In polynomial factorization, we often use the distributive property to check our work.
For example, with the factored polynomial 11y(5 - y^3), we apply the distributive property to expand it and verify our factorization:
  • Distribute 11y to 5: 11y * 5 = 55y
  • Distribute 11y to -y^3: 11y * -y^3 = -11y^4
Adding these gives us back the original polynomial:
55y - 11y^4
This verification ensures our factorization was done correctly and demonstrates the utility of the distributive property in solving polynomial equations.

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