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Chapter 5: Problem 66

Factor each polynomial. $$ 16 k^{2}+125 k^{5} $$

Short Answer

Expert verified
The factored form is $$ k^{2}(16 + 125 k^{3}) $$.

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, find the greatest common factor (GCF) of all the terms in the polynomial. In this case, both terms have a common factor of at least one power of k. The GCF is therefore $$ k^{2} $$.
02

Factor Out the GCF

Next, factor out the GCF from each term in the polynomial. Here, you factor out $$ k^{2} $$, resulting in: $$ 16 k^{2} + 125 k^{5} = k^{2}(16 + 125 k^{3}) $$.
03

Simplify the Inside Expression

After factoring out the GCF, analyze the remaining expression. Simplify if possible. In this case, the remaining expression $$16 + 125k^{3}$$ is already in its simplest form and cannot be factored further.

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Key Concepts

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

Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers or terms. In polynomial factorization, identifying the GCF is an essential step to simplify expressions.
To find the GCF of a polynomial, you need to:
  • Identify all the terms in the polynomial.
  • Determine the common factors of each term.
  • Choose the highest power of any common variables present.
For example, in the polynomial $$16 k^{2} + 125 k^{5}$$, both terms have at least one power of k in common. The GCF is therefore $$k^{2}$$.
Always remember to divide each term of the polynomial by the GCF. This helps in reducing the polynomial to its simplest form, making further calculations easier.
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors. It's like breaking down a complex expression into simpler pieces.
Here is a step-by-step guide to factor a polynomial:
  • Identify the GCF of the terms, as we discussed earlier.
  • Factor out the GCF from each term in the polynomial.
  • Write the polynomial as the product of the GCF and the remaining expression.
For example, let's factor the polynomial $$16 k^{2} + 125 k^{5}$$. We identified the GCF as $$ k^{2}$$. Next, we factor it out, resulting in: $$16 k^{2} + 125 k^{5} = k^{2}(16 + 125 k^{3})$$.
Now the polynomial is factored into $$k^{2}$$ and \(16 + 125k^{3}\).
Simplifying Expressions
Simplifying expressions involves reducing them to their most concise form while maintaining the same value.
After factoring out the GCF, the next step is to simplify the inner expression. In the example of $$16 k^{2} + 125 k^{5}$$, we factored it as $$k^{2}(16 + 125 k^{3})$$. Here, the inner expression $$16 + 125 k^{3}$$ cannot be simplified further, as there are no common factors or further factorable components.
The key steps in simplifying expressions are:
  • Factor out the GCF.
  • Analyze the remaining polynomial or expression.
  • Simplify any further possible factors.
If no further simplification is possible, you have completed the process. Simplifying polynomials makes them easier to work with, especially in advanced algebra and calculus.

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Most popular questions from this chapter

Factor each polynomial. $$ x^{6}-216 $$

Factor each polynomial. $$ 343 p^{3}+125 q^{3} $$

Factor each trinomial. \(r^{2}(r-s)-5 r s(s-r)-6 s^{2}(r-s)\)

Factor each polynomial. $$ y^{4}+y^{3}+y+1 $$

Skills In some cases, the method of factoring by grouping can be combined with the methods of special factoring discussed in this section. Consider this example. $$ \begin{array}{l} 8 x^{3}+4 x^{2}+27 y^{3}-9 y^{2} \\ \quad=\left(8 x^{3}+27 y^{3}\right)+\left(4 x^{2}-9 y^{2}\right) \\ \quad=(2 x+3 y)\left(4 x^{2}-6 x y+9 y^{2}\right)+(2 x+3 y)(2 x-3 y) \\ \quad=(2 x+3 y)\left[\left(4 x^{2}-6 x y+9 y^{2}\right)+(2 x-3 y)\right] \\ \quad=(2 x+3 y)\left(4 x^{2}-6 x y+9 y^{2}+2 x-3 y\right) \end{array} $$ Use the method just described to factor each polynomial. $$ 125 p^{3}+25 p^{2}+8 q^{3}-4 q^{2} $$
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