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Chapter 6: Problem 43

Write in factored form by factoring out the greatest common factor. \(65 y^{10}+35 y^{6}\)

Short Answer

Expert verified
5y^6(13y^4 + 7)

Step by step solution

01

- Identify the Greatest Common Factor

First, look at the coefficients of each term. For 65 and 35, the greatest common factor (GCF) is 5. Then consider the variable part. The lowest power of the common variable in both terms is the smallest exponent, which is y^6.
02

- Factor Out the GCF

Factor out the GCF, which is 5y^6, from each term in the expression. This means dividing each term by 5y^6 and writing what's left in parentheses.
03

- Write the Factored Expression

After factoring out the GCF from each term, we get: (5y^6)(13y^4 + 7). Thus, the expression in its factored form is: 5y^6(13y^4 + 7).

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Key Concepts

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

greatest common factor
When you are dealing with algebraic expressions, one fundamental concept is the greatest common factor (GCF). The GCF is the highest number that divides exactly into two or more numbers. When dealing with variables, it is the variable with the smallest exponent common to all terms. Identifying the GCF is crucial in simplifying expressions and factoring.

To find the GCF in the exercise given, follow these steps:
  • Look at the coefficients: For 65 and 35, the largest number that divides both is 5.
  • Consider the variable part: Both terms have the variable y, and the smallest exponent is 6, from y^6.

Hence, the GCF of the terms in the exercise is 5y^6.
factored form
Factored form is a way of expressing an algebraic expression as a product of its factors. Factoring helps in simplifying expressions and solving equations. To write an expression in factored form:
  • First, identify the GCF in all terms, as discussed.
  • Then, rewrite each term inside parentheses after factoring out the GCF. This means dividing each original term by the GCF.

In our specific example, after factoring out the GCF, 5y^6, from the expression 65y^{10} + 35y^6, we rewrite the expression inside parentheses: 13y^4 + 7. So, the factored form is (5y^6)(13y^4 + 7).
algebraic factoring
Algebraic factoring involves rewriting an expression as a product of its factors. This process commonly includes identifying the GCF and then factoring it out of the expression. Detailed steps include:
  • Identifying the GCF: First step, as explained, find the greatest common factor of the coefficients and the variables.
  • Rewriting the expression: Factor out the GCF from each term, which simplifies the original expression.
  • Writing the result: After factoring out the GCF, what remains is written in parentheses.

This process results in a simpler and often solvable expression. In our example, the algebraic factoring of 65y^{10} + 35y^6 yields the expression 5y^6(13y^4 + 7). Factoring helps in unlocking hidden solutions and is widely used in solving algebraic equations.

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Most popular questions from this chapter

Factor each trinomial completely. See Examples 1鈥�7. ( Hint: In Exercises 55鈥�58, first write the trinomial in descending powers and then factor.) $$ 15 n^{4}-39 n^{3}+18 n^{2} $$

Solve each equation, and check your solutions. $$ 6 x^{2}(2 x+3)=4(2 x+3)+5 x(2 x+3) $$

Find each product. \((x-3)(x-6)\)

Find the value of the indicated variable. Find \(b\) so that \(x^{2}+b x+25\) factors as \((x+5)^{2}\)

Factor each polynomial completely. $$ (a-b)^{3}-(a+b)^{3} $$
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