Problem 91 - Elementary Algebra

Question

For the following problems, expand the terms so that no exponents appear. $$ \left(x^{3}+7\right)^{2}\left(y^{2}-3\right)^{3}(z+10) $$

Step-by-Step Solution

Answer

Question: Expand the expression \((x^3 + 7)^2(y^2 - 3)^3(z + 10)\) Answer: The expanded expression is $x^6y^6z - 9x^6y^4z + 27x^6y^2z - 27x^6z + 10x^6y^6 - 90x^6y^4 + 270x^6y^2 - 270x^6 + 14x^3y^6z - 126x^3y^4z + 378x^3y^2z - 378x^3z + 140x^3y^6 - 1260x^3y^4 + 3780x^3y^2 - 3780x^3 + 49y^6z - 441y^4z + 1323y^2z - 1323z + 490y^6 - 4410y^4 + 13230y^2 - 13230$.

Step 1: Apply the binomial theorem to the first term

Apply the binomial theorem to the term \((x^3 + 7)^2\). Using the binomial theorem, we have: $$ (x^3 + 7)^2 = \binom{2}{0}(x^3)^2 + \binom{2}{1}(x^3)(7) + \binom{2}{2}7^2 $$ After simplifying, we get: $$ (x^3 + 7)^2 = x^6 + 14x^3 + 49 $$

Step 2: Apply the binomial theorem to the second term

Apply the binomial theorem to the term \((y^2 - 3)^3\). Using the binomial theorem, we have: $$ (y^2 - 3)^3 = \binom{3}{0}(y^2)^3 - \binom{3}{1}(y^2)^2(3) + \binom{3}{2}y^2(3)^2 - \binom{3}{3}3^3 $$ After simplifying, we get: $$ (y^2 - 3)^3 = y^6 - 9y^4 + 27y^2 - 27 $$

Step 3: Multiply all the terms

Multiply the expanded terms from steps 1 and 2, and the remaining term (z + 10) to get the final expanded form. We'll first multiply the expanded term from step 1 and the term (z+10): $$ (x^6 + 14x^3 + 49)(z + 10) = x^6z + 10x^6 + 14x^3z + 140x^3 + 49z + 490 $$ Now, multiply this result with the expanded term from step 2: $$ (x^6z + 10x^6 + 14x^3z + 140x^3 + 49z + 490)(y^6 - 9y^4 + 27y^2 - 27) $$ After multiplying, we get the final expanded form: $$ x^6y^6z - 9x^6y^4z + 27x^6y^2z - 27x^6z + 10x^6y^6 - 90x^6y^4 + 270x^6y^2 - 270x^6 + 14x^3y^6z - 126x^3y^4z + 378x^3y^2z - 378x^3z + 140x^3y^6 - 1260x^3y^4 + 3780x^3y^2 - 3780x^3 + 49y^6z - 441y^4z + 1323y^2z - 1323z + 490y^6 - 4410y^4 + 13230y^2 - 13230 $$

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental concept in algebra that provides a quick and efficient method for expanding expressions raised to a power, known as binomials. A binomial is a polynomial with exactly two terms, and when it's raised to a power, the binomial theorem describes how to expand it into a sum involving terms of the original binomial and binomial coefficients.

These coefficients follow a specific pattern, known as Pascal's Triangle, or can be calculated using combinatorial notation, \( \binom{n}{k} \), representing the number of ways to choose \( k \) elements from a set of \( n \) elements. The binomial theorem states:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
where \( a \) and \( b \) are any numbers, \( n \) is a non-negative integer, and the summation symbol \( \sum \) represents the sum of all terms obtained by taking the binomial coefficient multiplied by the respective powers of \( a \) and \( b \).

In the given exercise, the first step involves using the binomial theorem to expand \( (x^3 + 7)^2 \) by treating \( x^3 \) and \( 7 \) as \( a \) and \( b \) respectively, and \( 2 \) as the exponent \( n \).

Polynomial Expansion

Polynomial expansion is the process of simplifying an expression that involves polynomials so that it's written as a sum of terms without any exponents on the variables. This process utilizes algebraic operations including addition, subtraction, multiplication, and application of the binomial theorem. Expanding polynomials is a vital skill in algebra because it allows us to simplify complex expressions and solve equations.

For example, the exercise presents a polynomial expression that includes multiple binomials raised to powers. To expand these, one must systematically apply the binomial theorem to each term and then use multiplication to combine all the expanded terms together.

The exercise progresses by expanding each term step by step, first handling \( (x^3 + 7)^2 \) and \( (y^2 - 3)^3 \), and then combining the resultant expressions through multiplication. This multi-step approach breaks down the complex operation of expanding multiple binomials into more manageable, individual components, thereby avoiding confusion and simplifying the calculation.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power, which is the exponent indicating how many times to multiply the base by itself. It is denoted by the base followed by a superscript exponent. For instance, \( x^3 \) implies multiplying \( x \) by itself three times: \( x \cdot x \cdot x \).

In the context of the given exercise, the original expression contains several terms with exponents. Before the expansion, it is crucial to comprehend how to handle exponentiation when it appears in polynomial expressions. Specifically, when multiple polynomial terms, each raised to an exponent, are multiplied together, the result is obtained not just by multiplying the bases but also by properly applying the rules of exponentiation across the terms.

The final step of the exercise multiplies all the expanded terms together, which involves careful consideration of the order of operation and a systematic approach to ensure all terms are correctly accounted for. The exponentiation plays a key role in determining the final degree of each term in the expanded expression.