Problem 34 - Algebra for College Students

Question

Find each product. $$\left(\frac{3}{4} x\right)\left(-4 x^{2} y^{2}\right)\left(9 y^{3}\right)$$

Step-by-Step Solution

Answer

-27x^3y^5

Step 1: Multiply the Coefficients

First, identify and multiply the numerical coefficients from each term: \( \frac{3}{4}, -4, \) and \( 9 \). The product of these coefficients is: \( \frac{3}{4} \times -4 \times 9 = -27 \).

Step 2: Combine the \(x\) Terms

Next, multiply the \(x\) terms: In the expression, \( x \) and \( x^2 \) need to be combined by adding their exponents: \( x^1 \times x^2 = x^{1+2} = x^{3} \).

Step 3: Combine the \(y\) Terms

Now, combine the \(y\) terms. Notice that you have \( y^2 \times y^3 \) which can be combined by adding their exponents: \( y^{2+3} = y^5 \).

Step 4: Write the Complete Product

Combine all the results from the above steps: the coefficient \(-27\), the \(x\) term \(x^3\), and the \(y\) term \(y^5\). The complete product is \(-27x^{3}y^{5}\).

Key Concepts

Coefficients

When working with polynomial multiplication, understanding coefficients is crucial. Coefficients are the numerical or constant parts of the terms in an expression. In the original exercise: \( \left(\frac{3}{4} x\right)\left(-4 x^{2} y^{2}\right)\left(9 y^{3}\right) \), their coefficients are \( \frac{3}{4} \), \(-4\), and \(9\).
To find the product of these coefficients, multiply them together following ordinary multiplication rules: first multiply \( \frac{3}{4} \) by \(-4\), which results in \(-3\), and then multiply \(-3\) by \(9\) to get \(-27\).
By carefully multiplying these coefficients, you ensure the correct numerical answer, which is an important first step in polynomial multiplication.

Exponent Rules

Exponent rules are fundamental when dealing with variables raised to a certain power. In polynomial multiplication, these rules allow us to combine like terms. An exponent expresses how many times a number, known as the base, is multiplied by itself.
For example, in the expression \(x^1 \times x^2\), we add the exponents (1 + 2) to get \(x^3\). This is because the exponents indicate repeated multiplication of the base.
Similarly, for the \(y\) terms \(y^2\) and \(y^3\), add the exponents to obtain \(y^5\). Remember, exponents only combine when the bases are the same, so understanding these rules is key to combining polynomial expressions correctly.

Variable Combination

The combination of variables is an integral part of simplifying polynomial expressions. Different terms might have different variables such as \(x\) and \(y\), and each must be treated according to the rules of exponents.
First, group the like terms. Focus on the bases: combine the \(x\) bases \(x^1\) and \(x^2\) by adding their exponents to yield \(x^3\).
Second, do the same for the \(y\) terms; combine \(y^2\) and \(y^3\) to obtain \(y^5\).
Once like terms have been combined, it's easier to see the simplified form of the entire expression. Always ensure you have correctly matched and added the exponents for the bases to simplify an expression accurately. This skill will vastly improve your handling of polynomial multiplication.