Problem 42 - Elementary Algebra

Question

For the following problems, expand the quantities so that no exponents appear. $$ (a+10)^{2}\left(a^{2}+10\right)^{2} $$

Step-by-Step Solution

Answer

Answer: The fully expanded and simplified form of the expression is $a^{6}+20a^{5}+120a^{4}+400a^{3}+2200a^{2}+2000a+10000$.

Step 1: Expand the first term (a+10)^{2}

The first term is a binomial square, so we can use the formula (a+b)^{2}=a^{2}+2ab+b^{2}. In this case, a is 'a' and b is 10, so applying the formula, we get: $$ (a+10)^{2}=a^{2}+2(a)(10)+10^{2} $$

Step 2: Expand the second term (a^{2}+10)^{2}

The second term is also a binomial square, so we can use the same formula. Let a be 'a^{2}' and b be '10', then we get: $$ (a^{2}+10)^{2}=(a^{2})^{2}+2(a^{2})(10)+10^{2} $$

Step 3: Substitute the expansion of both terms

Now we can replace the original terms in the expression with their expansions: $$ (a^{2}+2(a)(10)+10^{2})((a^{2})^{2}+2(a^{2})(10)+10^{2}) $$

Step 4: Multiply the terms

Next, we will multiply the two expanded terms using the distributive property (FOIL method): $$ [(a^{2})(a^{4})]+[(a^{2})(20a^{2})]+[(a^{2})(100)]+[(20a)(a^{4})]+[(20a)(20a^{2})]+[(20a)(100)]+[(100)(a^{4})]+[(100)(20a^{2})]+[(100)(100)] $$

Step 5: Simplify the expression

Finally, we can simplify the expression by combining like-terms and simplifying the coefficients: $$ a^{6}+20a^{4}+100a^{2}+20a^{5}+400a^{3}+2000a+100a^{4}+2000a^{2}+10000 $$ This eliminates all of the exponents from the original expression.

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that describes the expansion of powers of a binomial—that is, a polynomial with two terms. Whenever students face a binomial raised to a power, such as \( (a + b)^n \), they can rely on this theorem for expansion. For instance, when expanding \( (a+10)^2 \) or \( (a^2+10)^2 \), we apply the binomial theorem, which provides a formula for each term in the expansion.

For a simple square of a binomial \( (a+b)^2 \), the theorem simplifies to \( a^2 + 2ab + b^2 \). This pattern is what has been applied in the provided solution to expand \( (a+10)^2 \) and \( (a^2+10)^2 \). The significance of understanding the binomial theorem lies in its general case, which allows expansion of any power of a binomial and, thus, is an invaluable tool in algebra.

Polynomial Expansion

Polynomial expansion involves writing out a polynomial that has been raised to a power in a form where no exponents remain, displaying all the individual terms. This process is critical when simplifying expressions or solving algebraic equations. The key to polynomial expansion is applying the distributive property repetitively over addition and multiplication.

In our exercise, we were asked to expand the quantity \( (a+10)^2(a^2+10)^2 \), which after applying the binomial theorem results in two expanded polynomials. Here, each term of the first expanded polynomial must be multiplied by each term of the second to get the final expanded form. Detailed knowledge of polynomial expansion allows students to handle more complex algebraic expressions and is a foundational skill in higher mathematics.

Distributive Property

The distributive property is a basic property of numbers that applies to both addition and multiplication. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then summing the products. In algebra, this property is what enables us to multiply two binomials—an operation commonly performed during polynomial expansion.

For example, when we have to multiply \( (a^2+2a.10+10^2) \) by \( (a^4+2a^2.10+10^2) \) as in Step 4 of the solution, we use the distributive property to ensure each term in the first set multiplies every term in the second set. This approach is foundational in algebra and helps in simplifying expressions to their most basic form, facilitating further operations like solving equations or graphing polynomial functions.

FOIL Method

The FOIL method is a specific application of the distributive property used to multiply two binomials. FOIL is an acronym for First, Outer, Inner, Last, referring to the terms in each binomial. This method is particularly useful when dealing with quadratic expressions or any time you expand binomials.

Using the FOIL method, you would first multiply the First terms of each binomial, then the Outer terms, followed by the Inner terms, and finally, the Last terms. This method comes into play in our Step 4, ensuring a systematic approach to multiplying the expanded forms of the binomials. While the FOIL method deals with two-binomial multiplication, it sets the stage for understanding more advanced polynomial multiplication, where a similar distributive process is used repeatedly.