Problem 72 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ -a^{2} b^{3}\left(6 a b^{4}+5 a b^{3}-8 b^{2}+7 b-2\right) $$

Step-by-Step Solution

Answer

Question: Multiply the given monomial with the polynomial and simplify: (-a^2 b^3)(6 a b^4 + 5 a b^3 - 8 b^2 + 7 b - 2) Answer: -6a^3b^7 - 5a^3b^6 + 8a^2b^5 - 7a^2b^4 + 2a^2b^3

Step 1: Multiply the monomial with each term of the polynomial

First, we need to multiply the given monomial (-a^2 b^3) with each term of the polynomial inside the parentheses: $$ (-a^{2}b^{3})\left(6 a b^{4}+5 a b^{3}-8 b^{2}+7 b-2\right) $$

Step 2: Apply Distributive Property

Using the distributive property, we will multiply the monomial with each term of the polynomial: $$ (-a^{2}b^{3})(6 a b^{4})+(-a^{2}b^{3})(5 a b^{3})+(-a^{2}b^{3})(-8 b^{2})+(-a^{2}b^{3})(7 b)+(-a^{2}b^{3})(-2) $$

Step 3: Apply the rules of exponents and simplify

Now we will apply the rules of exponents, that is, when we multiply terms with the same base, we need to add the exponents: $$ -6a^{3}b^{7}-5a^{3}b^{6}+8a^{2}b^{5}-7a^{2}b^{4}+2a^{2}b^{3} $$

Step 4: Check for any like terms

After expanding the expression, we can analyze that there are no like terms as no terms have the same variables and powers.

Step 5: Write the final answer

Our final simplified expression is: $$ -6a^{3}b^{7}-5a^{3}b^{6}+8a^{2}b^{5}-7a^{2}b^{4}+2a^{2}b^{3} $$

Key Concepts

Distributive Property

Understanding the distributive property is critical when it comes to polynomial multiplication. It's like sharing something equally among friends. In mathematical terms, when you have a monomial outside the parentheses and a polynomial inside, such as \( -a^{2} b^{3}\left(6 a b^{4}+5 a b^{3}-8 b^{2}+7 b-2\right) \), you distribute the outside monomial to EVERY term within the polynomial. Picture each term inside as a guest waiting for a slice of the monomial pie. So you take \( -a^{2} b^{3} \) and multiply it by each term, one by one, ensuring that each term gets an equal 'share' of the monomial. The result will be a series of new terms that have been affected by the monomial, setting the stage for further simplification.

Combining Like Terms

After distribution, we might end up with a cluttered expression full of various terms. It's like a room scattered with toys—some of them are similar and need to be grouped together. That's what combining like terms is all about. Look at the result of the distribution; are there any terms with the exact same variables raised to the same powers? Those are 'like terms', and they can be combined—just like cleaning up a room by grouping similar toys together.

Add or subtract them as appropriate to simplify the expression. However, just as not all toys are similar, sometimes, after multiplication, no terms match. In our example \( -6a^{3}b^{7}-5a^{3}b^{6}+8a^{2}b^{5}-7a^{2}b^{4}+2a^{2}b^{3} \), there are no like terms, so the expression remains as it is—neat and tidy from the start!

Rules of Exponents

When multiplying polynomials, the rules of exponents are the guide to simplifying expressions. These rules are the roadmap for navigating through an otherwise complex landscape of variables and numbers. Remember, when you multiply two terms with the same base, you keep the base and add their exponents. For instance, multiplying \(a^{2}\) by \(a^{3}\) gives \(a^{2+3} = a^{5}\), not \(a^{6}\). It’s akin to stacking blocks of the same color; you build up, not out.

In the exercise \( -a^{2} b^{3}\left(6 a b^{4}\right) \) becomes \( -6a^{2+1}b^{3+4} \) following this rule. Ensuring that these rules are correctly applied is essential for arriving at the simplified expression.

Monomial Multiplication

When it comes to multiplying a monomial by a polynomial, think of it as repetitively using a stamp. Each time you press the stamp (the monomial), you leave its mark on a sheet of paper (a term of the polynomial). Multiplying a monomial by each term in a polynomial involves repeating this process as many times as there are terms. This individual term-by-term multiplication might sound laborious, but it's the mechanical heart of polynomial multiplication.

Every time you multiply a monomial by a term, you apply the exponent rules and simplify. The product is the accumulation of these simple steps. For example, \( -a^{2} b^{3} \) multiplied by \( 5ab^{3} \) results in \( -5a^{2+1} b^{3+3} \) or \( -5a^{3}b^{6} \)—just as a stamp leaves a clear and precise imprint on paper.