Problem 39 - Algebra Form and Function

Question

Without expanding, what is the constant term of $$ (x+2)(x+3)(x+4)(x+5)(x+6) ? $$

Step-by-Step Solution

Answer

Answer: The constant term of the given expression without expanding is \(720\).

Step 1: Identify the constant term in each factor.

In the given expression, there are five factors. Let us identify the constant term in each factor: 1. \((x + 2)\): The constant term is \(2\). 2. \((x + 3)\): The constant term is \(3\). 3. \((x + 4)\): The constant term is \(4\). 4. \((x + 5)\): The constant term is \(5\). 5. \((x + 6)\): The constant term is \(6\).

Step 2: Find the product of the constant terms.

Now, we will find the product of the constant terms from each factor: $$ 2 \times 3 \times 4 \times 5 \times 6 $$

Step 3: Calculate the product.

We will now calculate the product of the constant terms: $$ 2 \times 3 \times 4 \times 5 \times 6 = 720 $$

Step 4: Conclusion

The constant term of the given expression, without expanding, is \(720\).

Key Concepts

Polynomial Multiplication

When dealing with polynomials, multiplication is a method used to combine two or more polynomial expressions into one. Each polynomial is generally expressed as a sum of terms that include variable powers and coefficients. During multiplication, each term in the first polynomial is multiplied by each term in the second polynomial. The process also involves adding like terms together once all multiplications are done.

In the expression \((x+2)(x+3)(x+4)(x+5)(x+6)\), polynomial multiplication would typically involve expanding the entire expression to find all resultant terms. However, for some exercises such as finding the constant term, you can skip full expansion by focusing on particular parts of the multiplication. This allows us to quickly reach the answer without unnecessary calculations.

Constant Term Analysis

To understand constant term analysis, it's essential to recognize what a constant term is. In any polynomial, the constant term is the part that does not contain any variable. This is the constant whose value remains the same for all values of the variable.

In the context of polynomial multiplication, as seen in \((x+2)(x+3)(x+4)(x+5)(x+6)\), the constant term in each polynomial factor comprises its numeric standalone part:

• For \((x+2)\), the constant is 2.
• For \((x+3)\), the constant is 3.
• Continuing this way, the constants are 4, 5, and 6 for the remaining factors respectively.

Calculating the product of these constant values gives the constant term for the entire expression without expansion. So, in our example, the constant term is achieved by multiplying these constants: \(2 \times 3 \times 4 \times 5 \times 6 = 720\).

Thus, the constant term of the original polynomial is 720.

Factored Form of Polynomials

When a polynomial is expressed as a product of simpler polynomials, it is said to be in its factored form. Factored form helps in simplifying polynomial expressions and in solving polynomial equations. Additionally, it's particularly useful for identifying roots of the polynomial and analyzing specific terms.

The expression \((x+2)(x+3)(x+4)(x+5)(x+6)\) is a factored form representation of a polynomial. Instead of expanding it to a standard polynomial form, which would involve tedious multiplying and adding of terms, keeping it in factored form allows for easier calculation of certain polynomial features like the constant term.

• Factored form is efficient in computations concerning specific terms because each algebraic expression can be tackled independently.
• It showcases the utility of focusing on specific coefficients, such as the constant term, without full expansion of the polynomial.

This structured form benefits not only in theoretical mathematics but in practical situations where simplifying calculations is necessary.