Problem 75 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$(2-3 x)(5-2 x)\left(x^{2}-1\right)$$

Step-by-Step Solution

Answer

The product is \( 6x^4 - 19x^3 + 4x^2 + 19x - 10 \).

Step 1: Distribute the First Two Binomials

The expression is \( (2 - 3x)(5 - 2x)(x^2 - 1) \). Start by multiplying the first two binomials \( (2 - 3x)(5 - 2x) \). Distribute each term from \( (2 - 3x) \) across each term in \( (5 - 2x) \): \[ \begin{align*} (2)(5) + (2)(-2x) + (-3x)(5) + (-3x)(-2x) \end{align*} \]This simplifies to \( 10 - 4x - 15x + 6x^2 \). Combine the like terms: \( 10 - 19x + 6x^2 \).

Step 2: Multiply by the Third Polynomial

Now multiply the result from Step 1, \( 6x^2 - 19x + 10 \), by the third polynomial \( (x^2 - 1) \):\[ \begin{align*} (6x^2 - 19x + 10)(x^2 - 1) \end{align*} \]Distribute each term in \( (6x^2 - 19x + 10) \) across each term in \( (x^2 - 1) \):\[ \begin{align*} (6x^2)(x^2) + (6x^2)(-1) + (-19x)(x^2) + (-19x)(-1) + (10)(x^2) + (10)(-1) \end{align*} \]This expands to \( 6x^4 - 6x^2 - 19x^3 + 19x + 10x^2 - 10 \).

Step 3: Combine Like Terms

Combine the like terms from the expanded expression: \[ \begin{align*} 6x^4 - 19x^3 + (10x^2 - 6x^2) + 19x - 10 \end{align*} \]This results in \( 6x^4 - 19x^3 + 4x^2 + 19x - 10 \).

Step 4: Conclusion

The final expanded form after multiplying all the polynomials is \( 6x^4 - 19x^3 + 4x^2 + 19x - 10 \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions and perform multiplication more efficiently. It allows us to distribute a multiplication operation over addition or subtraction inside parentheses. This means that when you have an expression like \((a + b)(c + d)\), you can multiply each term in the first parenthesis by each term in the second parenthesis:

• \(a \times c\)
• \(a \times d\)
• \(b \times c\)
• \(b \times d\)

All these products are then added together. The same process applies when multiplying binomials or any polynomials together. In the originał exercise, each term from one binomial was distributed and multiplied with every term in the other binomial.

Combining Like Terms

After using the distributive property and expanding all the terms, we often end up with many terms that can be combined. These are called 'like terms.'

• Like terms are terms that have the same variable raised to the same power.
• If the terms only differ by their coefficients, they can be combined.

For example, in the expression \(10 - 4x - 15x + 6x^2\), the terms \(-4x\) and \(-15x\) are like terms, because both contain the variable \(x\) raised to the power of 1. Combining them gives \(-19x\). This simplification step is crucial for obtaining a clearer and more manageable expression after polynomial multiplication.

Binomials

A binomial is a polynomial with exactly two terms. These terms are typically separated by a plus or minus sign.

• An example of a binomial is \(a + b\) or \(3x - 4\).

When dealing with polynomial multiplication involving binomials, we often use the distributive property twice, as seen in \((2 - 3x)(5 - 2x)\). Each term in the first binomial is multiplied by each term in the second, resulting in four terms that need to be simplified in further steps. Understanding how to handle binomials effectively is key to working with more complex polynomial expressions.

Polynomial Expansion

Polynomial expansion involves spreading out a polynomial expression so that it's written as a sum of terms without any parentheses. This process allows us to work with polynomials more easily.

• Each step involves multiplying terms and then combining like terms to simplify the expression.

For instance, starting from an expression like \((6x^2 - 19x + 10)(x^2 - 1)\), we separately find each product using distribution. Once we expand it all, we end up with a longer polynomial like \(6x^4 - 19x^3 + 4x^2 + 19x - 10\). Through polynomial expansion, complex and nested expressions are unfolded into simpler, linear combinations. This makes further analysis and utilization, such as solving or graphing, more straightforward.