Problem 63 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$\left(2 x^{2}+3 x-4\right)\left(x^{2}-2 x-1\right)$$

Step-by-Step Solution

Answer

The product is \(2x^4 - x^3 - 12x^2 + 5x + 4\).

Step 1: Apply Distributive Property

To find the product of the two polynomials \((2x^2 + 3x - 4)(x^2 - 2x - 1)\), we will use the distributive property by multiplying each term in the first polynomial by each term in the second polynomial.

Step 2: Multiply Each Term Individually

Start with the first term from the first polynomial \(2x^2\) and multiply it by each term in \((x^2 - 2x - 1)\): 1. \(2x^2 \cdot x^2 = 2x^4\)2. \(2x^2 \cdot (-2x) = -4x^3\)3. \(2x^2 \cdot (-1) = -2x^2\)Next, take the second term \(3x\) and multiply it by each term in the second polynomial:1. \(3x \cdot x^2 = 3x^3\)2. \(3x \cdot (-2x) = -6x^2\)3. \(3x \cdot (-1) = -3x\)Finally, take \(-4\) and multiply it by each term in the second polynomial:1. \(-4 \cdot x^2 = -4x^2\)2. \(-4 \cdot (-2x) = 8x\)3. \(-4 \cdot (-1) = 4\)

Step 3: Combine Like Terms

Combine all the terms obtained from the multiplication:\[2x^4 - 4x^3 + 3x^3 - 2x^2 - 6x^2 - 4x^2 - 3x + 8x + 4\]Combine like terms:- For \(x^4\): \(2x^4\)- For \(x^3\): \(-4x^3 + 3x^3 = -x^3\)- For \(x^2\): \(-2x^2 - 6x^2 - 4x^2 = -12x^2\)- For \(x\): \(-3x + 8x = 5x\)- Constant: \(4\)Therefore, the combined expression is \(2x^4 - x^3 - 12x^2 + 5x + 4\).

Step 4: Write the Final Expression

After combining all the like terms, the final expanded expression is \[2x^4 - x^3 - 12x^2 + 5x + 4\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a term across a set of terms within parentheses. It’s a crucial tool for multiplying polynomials effectively. In our example with the polynomials \((2x^2 + 3x - 4)\) and \((x^2 - 2x - 1)\), we use the distributive property to simplify the multiplication process.

By applying the distributive property, each term from the first polynomial is multiplied by every term in the second polynomial. This ensures that all combinations of terms are considered. For instance:

• Multiply the first term from the first polynomial \(2x^2\) by every term in the second polynomial: \(x^2, -2x, \) and \(-1\).
• Repeat the process for the second term \(3x\) and third term \(-4\).

This systematic approach helps us to handle more complex polynomial multiplications with ease, breaking them down into manageable parts.

Combining Like Terms

After applying the distributive property, we often end up with a long list of terms of various powers. At this point, it is essential to combine like terms to simplify the expression.

Like terms are terms that have the same variables raised to the same power. They can be combined by simply adding or subtracting their coefficients. For example, if you have terms like \(-4x^3\) and \(3x^3\), they are both \(x^3\) terms, so you can combine them: \(-4x^3 + 3x^3 = -x^3\).

• Identify terms with the same exponent and variable, like \(x^3\), \(x^2\), etc.
• Add or subtract their coefficients according to the expression.

This process helps condense the expression and gives you a clearer, simpler polynomial result, such as our final result of \(2x^4 - x^3 - 12x^2 + 5x + 4\).

Binomial Multiplication

Binomial multiplication refers to the process of multiplying two binomials, which are polynomials that contain exactly two terms each. Although our original problem involves trinomials rather than binomials, understanding binomial multiplication is key to tackling polynomials of any length.

When multiplying two binomials, follow these steps:

• Use the distributive property to multiply each term in the first binomial by each term in the second binomial.
• Combine like terms to simplify the expression.

The FOIL method (First, Outer, Inner, Last) is often used for binomials: multiply the First terms, then the Outer terms, then Inner, and finally Last terms. Though FOIL doesn't apply directly to trinomials or larger polynomials, mastering it aids in understanding larger polynomial multiplications.

Special Polynomial Patterns

Special polynomial patterns can simplify the multiplication process when recognized. Common patterns include the difference of squares and perfect square trinomials.

These patterns are shortcuts that can help you multiply certain polynomials more quickly. For example:

• Difference of Squares: If you see a pattern like \(a^2 - b^2\), it can be rewritten as \((a+b)(a-b)\).
• Perfect Square Trinomial: A trinomial like \((a+b)^2 = a^2 + 2ab + b^2\) is the result of squaring a binomial.

While the exercise involved trinomials, being able to spot these patterns can save you time and effort by allowing you to apply these shortcuts when applicable in other problems.