Problem 56 - Elementary Algebra

Question

Find the product. \(x^{2}(x-3)(x+4)\).

Step-by-Step Solution

Answer

Answer: The product of these expressions is \(x^{4} + x^{3} - 12x^{2}\).

Step 1: Multiply the first two expressions (\((x^2)(x-3)\))

Apply the distributive property by multiplying each term in the second expression by the first expression \((x^2)\): \((x^2)(x) - (x^2)(3)\). Perform the multiplication: \(x^{3} - 3x^2\).

Step 2: Multiply the expression from step 1 by the third expression (\([x^{3}-3x^2](x+4)\))

Apply the distributive property by multiplying each term in the expression from step 1 by each term in the third expression: \((x^3)(x) + (x^3)(4) - (3x^2)(x) - (3x^2)(4)\). Perform the multiplication: \(x^4 + 4x^3 - 3x^3 - 12x^2\).

Step 3: Combine like-terms

In the expression from step 2, there are two like terms (\(4x^3\) and \(-3x^3\)). Combine these terms: \(x^4 + 1x^3 -12x^2\).

Step 4: Write the final product

The product of the given expression \(x^{2}(x-3)(x+4)\) is \(x^{4} + x^{3} - 12x^{2}\).

Key Concepts

Distributive Property

The distributive property is a crucial concept in algebra that allows us to simplify expressions and solve equations. It involves distributing, or spreading out, one term over the terms inside a parentheses. Essentially, it means multiplying a single term by each term within a set of parentheses. This property is written in the form:

• If you have an expression like \(a(b + c)\), it can be expanded to \(ab + ac\).

To apply this to polynomials, picture multiplying \(x^2\) by \((x - 3)\). You take \(x^2\) and apply it to both \(x\) and \(-3\). It becomes \((x^2) \cdot x - (x^2) \cdot 3\), which simplifies to \(x^3 - 3x^2\).
Then by engaging with the third polynomial, \((x + 4)\), each term from the result \(x^3-3x^2\) is distributed once more. This means multiplying each term by every term of \((x+4)\): \(x^3 \cdot x\), \(x^3 \cdot 4\), \(-3x^2 \cdot x\), and \(-3x^2 \cdot 4\). The step-by-step application of this property simplifies the polynomial multiplication process.

Combining Like Terms

After applying the distributive property, we often face the task of combining like terms. Like terms are those that have the same variable raised to the same power, but may have different coefficients.
By combining them, we can simplify polynomials further. The process involves adding or subtracting the coefficients of terms that are alike.

• For instance, in our expression from the step-by-step solution, \(4x^3\) and \(-3x^3\) are like terms. Notice that both have the base variable \(x\) raised to the power of three.

You simply combine these by adding their coefficients: \(4 - 3\), which results in \(1x^3\) or simply \(x^3\).
This process helps transform the expanded polynomial into a simpler, cleaner expression: \(x^4 + x^3 - 12x^2\). Always ensuring to combine like terms can prevent errors and make it easier to work with polynomials in algebra.

Polynomials

Polynomials are algebraic expressions that involve variables raised to whole number exponents. These take the form of sums and differences of terms such as \(x^2\), \(2x\), or constants like \(4\). Each term in a polynomial is a combination of a variable and a coefficient.
An example polynomial might look like \(3x^2 + 2x - 5\).
Here are some key characteristics of polynomials:

• A single variable polynomial like \(x^{2}(x-3)(x+4)\) can be broken down into simpler parts through multiplication and simplification.
• The degree of a polynomial is the highest power of the variable in the expression. For the final expression \(x^4 + x^3 - 12x^2\), the degree is 4 due to the highest term \(x^4\).
• Polynomials are the building blocks for more complex algebra concepts and play a key role in calculus, statistics, and higher mathematics.

Understanding how to handle and manipulate polynomials using multiplication and simplification techniques empowers students to tackle more complex problems in mathematics confidently.