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Chapter 10: Problem 26

Determine the greatest common factor. \(15 c^{3} d^{3}\) and \(10 c^{4} d^{2}\)

Short Answer

Expert verified
The greatest common factor is 5c^{3}d^{2}.

Step by step solution

01

Identify Prime Factors of Coefficients

Break down the coefficients of each term into their prime factors. For 15, the prime factors are 3 and 5. For 10, the prime factors are 2 and 5.
02

Find Common Prime Factors

Identify the prime factors that are common to both 15 and 10. The common prime factor is 5.
03

Identify Lowest Power of Each Variable

For the variables, compare the exponents in each term. For the variable 'c', the powers are 3 and 4. So, the lowest power is 3. For the variable 'd', the powers are 3 and 2. So, the lowest power is 2.
04

Combine Resulting Factors

Combine the common prime factor with the variables using their lowest powers: The greatest common factor is therefore 5c^{3}d^{2}.

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Key Concepts

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

Prime Factors
Understanding prime factors is a key step in determining the greatest common factor (GCF). Prime factors are the prime numbers that multiply together to give the original number. Let's look at the coefficients in the exercise: 15 and 10.

To find the prime factors:
  • For 15, the prime factors are 3 and 5 since 3 x 5 = 15.
  • For 10, the prime factors are 2 and 5 because 2 x 5 = 10.

The common prime factor between 15 and 10 is 5. This common factor will help us find the GCF of the given terms.
Coefficients in Algebra
Coefficients in algebra are the numerical parts of terms. In the exercise, the coefficients are 15 and 10. To find the GCF, we first break these down into their prime factors, which we've discussed above.

Next, we look at the variables: in this case, 'c' and 'd'. Each variable has an exponent, indicating how many times that variable is multiplied by itself. For the GCF, we choose the smallest exponent for each variable:
  • For 'c', we have 15c3d3 and 10c4d2. The smallest exponent for 'c' is 3.
  • For 'd', the smallest exponent is 2.

So, our coefficients now include the factors 5 (common prime factor), c3, and d2.
Exponents
Exponents are used to indicate how many times a number or variable is multiplied by itself. They play a crucial role when determining the GCF in algebraic expressions.

In our exercise, we have terms with exponents: 15c3d3 and 10c4d2. The GCF involves taking the smallest exponent for each variable:
  • For 'c', compare the exponents: 3 and 4. The lowest is 3.
  • For 'd', the exponents are 3 and 2. The lowest is 2.
Combining the lowest exponents with the common prime factor 5, we get the GCF: 5c3d2.Understanding exponents ensures you accurately find the GCF for expressions with variables.

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Most popular questions from this chapter

Fill in the missing polynomial. $$(\quad)(z-5)=z^{2}-3 z-10$$

Simplify. Write the answers with positive exponents only. $$\frac{4^{2}}{4^{-1}}$$

Simplify the expression. $$\left(2 x y^{4}\right)^{3}\left(4 x^{2} y^{3}\right)$$

Subtract the polynomials. $$\left(-8 y^{4}+2 y^{2}-3 y+10\right)-\left(6 y^{4}+y^{3}+11 y-9\right)$$

Subtract the polynomials. $$\begin{array}{r}15 a^{3}-2 a^{2}+4 a \\\\-\left(2 a^{3}+6 a^{2}-12 a\right) \\\\\hline\end{array}$$
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