Problem 88 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ \left(4 a^{2} b^{3}-2 a\right)\left(5 a^{2} b-3 b\right) $$

Step-by-Step Solution

Answer

Question: Multiply the expressions $(4a^2b^3 - 2a)(5a^2b - 3b)$ and simplify the resulting expression. Answer: The simplified expression after multiplying the given expressions is $20a^4b^4 - 12a^2b^4 - 10a^3b + 6ab$.

Step 1: Apply the distributive property

To perform the multiplication, we will apply the distributive property. This means that we will multiply each term in the first expression by each term in the second expression: $$ (4a^2b^3 - 2a)(5a^2b - 3b) = (4a^2b^3 \times 5a^2b) - (4a^2b^3 \times 3b) - (2a \times 5a^2b) + (2a \times 3b) $$

Step 2: Simplify each term

Now, we need to simplify the terms obtained in Step 1 by combining the variables and constants: $$ = (4 \times 5a^2a^2b^3b ) - (4 \times 3a^2b^3b) - (2 \times 5a^3b) + (2 \times 3ab) $$

Step 3: Rewrite the exponents

Next, rewrite the exponents by adding them for the same bases in each term: $$ = (20a^{2+2}b^{3+1}) - (12a^{2}b^{3+1}) - (10a^{1+2}b) + (6a^{1}b) $$

Step 4: Calculate the exponents

Now, calculate the sum of the exponents for the same bases in each term: $$ = (20a^4b^4) - (12a^{2}b^4) - (10a^3b) + (6ab) $$

Step 5: Check for like terms

Inspect the expression for any like terms. We can see that there are no like terms in this expression: $$ 20a^4b^4 - 12a^2b^4 - 10a^3b + 6ab $$ Since there are no like terms in this expression, our final answer is: $$ 20a^4b^4 - 12a^2b^4 - 10a^3b + 6ab $$

Key Concepts

distributive property

When you encounter polynomial multiplication, the distributive property is an essential tool. This property allows you to multiply two binomials by distributing each term of one polynomial across the terms of the other. Think of it as spreading each term from the first polynomial over each term in the second.
For example, with the expression \((4a^2b^3 - 2a)(5a^2b - 3b)\), apply the distributive property by multiplying each component separately. This means writing it out like this:
* \(4a^2b^3 \times 5a^2b\)* \(4a^2b^3 \times -3b\)* \(-2a \times 5a^2b\)* \(-2a \times -3b\)
Remember, each term from the first polynomial affects each term in the second. This step sets up the expressions for further simplification.

combining like terms

Once you've used the distributive property to expand your expression, it's time to simplify by combining like terms. Like terms in a polynomial share the same variable raised to the same power. For example, \(3x^2\) and \(5x^2\) are like terms because they both contain \(x^2\).
In our exercise, after distributing the terms, we look for similar terms that we can combine. Here, even after distributing, no terms share both the same variables and powers. This means our expression ends as:
* \(20a^4b^4\)* \(- 12a^2b^4\)* \(- 10a^3b\)* \(+ 6ab\)
Always look for the terms that can be grouped together, but in this case, there are none. If terms could be combined, it would involve adding or subtracting their coefficients.

exponent rules

Exponent rules are a crucial part of dealing with polynomial expressions, particularly when multiplying terms. When multiplying powers with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\). This rule keeps calculations consistent and simplifies the expression accurately.
In our example, each time you multiplied the terms like \(4a^2b^3 \times 5a^2b\), you apply the rule: * Multiply the coefficients: \(4 \times 5 = 20\)* Add the exponents of \(a\): \(a^{2+2} = a^4\)* Similarly, add the exponents of \(b\): \(b^{3+1} = b^4\)
Understanding how to rewrite exponents by adding them together ensures that every term is expressed correctly, leading to accurate polynomial expressions. Always check that bases are the same before adding exponents!