Problem 19 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ -60 x^{5} b^{3} f^{9}, \quad-15 x^{2} b^{2} f^{2} $$

Step-by-Step Solution

Answer

Answer: The other factor is \(4x^3bf^7\).

Step 1: Identify the product and given factor

The product is given as \(-60x^{5}b^{3}f^{9}\) and the given factor is \(-15x^{2}b^{2}f^{2}\).

Step 2: Divide the product by the given factor

To find the other factor, we should divide the given product by the factor. $$ \frac{-60x^{5}b^{3}f^{9}}{-15x^{2}b^{2}f^{2}} $$

Step 3: Divide the numerical coefficients

We will start by dividing the numerical coefficients (\(-60\) and \(-15\)). $$ \frac{-60}{-15}=4 $$

Step 4: Divide the x terms

Now we'll divide the x terms (\(x^5\) and \(x^2\)). $$ \frac{x^5}{x^2}=x^{5-2}=x^3 $$

Step 5: Divide the b terms

Next, we'll divide the b terms (\(b^3\) and \(b^2\)). $$ \frac{b^3}{b^2}=b^{3-2}=b^1=b $$

Step 6: Divide the f terms

Finally, we'll divide the f terms (\(f^9\) and \(f^2\)). $$ \frac{f^9}{f^2}=f^{9-2}=f^7 $$

Step 7: Combine the results from Steps 3-6

Now, we'll combine the results from the previous steps to find the other factor. $$ 4\cdot x^3\cdot b\cdot f^7 = 4x^3bf^7 $$ The other factor of the given product is \(4x^3bf^7\).

Key Concepts

Polynomial Division

Polynomial division is much like long division with numbers, but instead of numbers, we use terms with variables and exponents. It helps us find out what we multiply with one expression to get another one. When dividing polynomials, we focus on dividing each term separately.

• First, divide the coefficients (the constant numbers in front of the variables). This means performing regular number division.
• Then, look at the variables which might have exponents attached. We subtract the exponents of similar variables in the divisor from those in the dividend to find the new exponent.
• We repeat this process for each term in the expression to get the final answer.

A crucial part is to align like terms, ensuring we only divide terms with the same bases.

Exponents

Exponents signify repeated multiplication of a base number. In algebra, understanding how to handle exponents is vital for dividing polynomials. Let's dissect how they work.

• Exponents show how many times to use the base in a multiplication. For example, in "\( x^5 \)", "\( x \)" is the base and "5" is the exponent, so it means "\( x \times x \times x \times x \times x \)".
• When dividing terms with the same base, we subtract their exponents. For example, \( \frac{x^5}{x^2} = x^{5-2} = x^3 \).
• This rule applies to other variables such as "\( b \)" and "\( f \)" in our polynomial division, simplifying the expressions accordingly.

By mastering how to deal with exponents, dividing terms becomes straightforward and systematic.

Factorization

Factorization involves breaking down a complex expression into simpler "factors" that, when multiplied together, can give back the original expression. This is a fundamental concept in algebra. In the given problem, one factor is already provided, and factorization helps identify the missing component. Here’s how we approach it:

• The primary idea is to determine which whole expressions multiply together to recreate the product.
• In polynomial division, after dividing by a known factor, the result is another factor of the original product.
• Recombine the known factor with the newly determined one to reassess their multiplication against the original expression.

Ensure accuracy by verifying if the multiplication of identified factors results back in the original polynomial.