Problem 71 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 9 y^{3}\left(2 y^{4}-3 y^{3}+8 y^{2}+y-6\right) $$

Step-by-Step Solution

Answer

Question: Multiply and simplify the expression 9y^3(2y^4 - 3y^3 + 8y^2 + y - 6). Answer: 18y^7 -27y^6 +72y^5 +9y^4 -54y^3

Step 1: Distribute 9y^3 to each term inside the parentheses

We will multiply 9y^3 by each term in the parentheses one by one: $$ 9y^3 \times 2y^4 = 18y^{7} $$ $$ 9y^3 \times -3y^3 = -27y^{6} $$ $$ 9y^3 \times 8y^2 = 72y^{5} $$ $$ 9y^3 \times y = 9y^{4} $$ $$ 9y^3 \times -6 = -54y^{3} $$ Next, write the expression with the resulting terms: $$ 18y^7 -27y^6 +72y^5 +9y^4 -54y^3 $$

Step 2: Combine any like terms

Since there are no like terms in the expression, we don't need to combine any terms. So our final answer is: $$ 18y^7 -27y^6 +72y^5 +9y^4 -54y^3 $$

Key Concepts

Combining Like Terms

When we work with algebraic expressions, identifying and combining like terms is essential for simplifying expressions. Like terms are terms that have the exact same variable parts, which means they have the same variables raised to the same powers. For example, 3x^2 and 5x^2 are like terms, but 3x^2 and 3x^3 are not.

To combine like terms, we simply add or subtract their coefficients, the numerical part in front of the variables. For instance, if we have 3x^2 + 5x^2, we combine these to yield 8x^2. On the flip side, if no terms share the same variable part, like in our original exercise with 18y^7, -27y^6, 72y^5, 9y^4, and -54y^3, there's nothing to combine. Each term is unique in its variables and exponents, so the expression is already as simplified as it can be.

A common pitfall for students is trying to combine terms that are not like terms, which can result in incorrect simplification. It's vital to double-check variables and their exponents to avoid this mistake.

Distributive Property

The distributive property is a cornerstone of algebra that allows us to multiply a single term by each term within a parenthesis, thus 'distributing' the multiplier. This property can be written algebraically as a(b + c) = ab + ac. When applied, we multiply the term outside the parentheses (in this case, a) with each term inside the parentheses (b and c), then sum up the results.

This property not only applies to simple numbers but also extends to more complex expressions involving variables and exponents, such as the one from our exercise. Here, the term 9y^3 was distributed across the polynomial inside the parentheses, leading to the multiplication of 9y^3 with each term 2y^4, -3y^3, 8y^2, y, and -6. Understanding and correctly applying the distributive property is crucial for polynomial multiplication and higher-level algebra.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They can range from simple expressions like 3x + 4 to more complex polynomials like the one in our exercise.

An important skill in dealing with algebraic expressions is recognizing their structure and knowing which operation to perform first. To avoid confusion, especially in longer expressions, we apply the order of operations: parentheses first, followed by exponents, then multiplication and division, and lastly, addition and subtraction. Expressions can be manipulated using properties of operations, such as the distributive property, or by combining like terms, to arrive at a simplified form or solve for a variable.

Mastery of algebraic expressions is essential for progress in mathematics, as it forms the basis for equations, inequalities, functions, and countless application problems in advanced topics.