Problem 78 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$(4+x)\left(2 x^{2}-3\right)$$

Step-by-Step Solution

Answer

The product is \(2x^3 + 8x^2 - 3x - 12\).

Step 1: Apply the Distributive Property

To multiply the polynomials \((4+x)\) and \((2x^2-3)\), we apply the distributive property. This means we need to multiply each term in the first polynomial by each term in the second polynomial. Start with multiplying \(4\) by each term in \((2x^2-3)\).

Step 2: Multiply First Monomial

Multiply \(4\) by each term in \((2x^2-3)\): - \(4 \cdot 2x^2 = 8x^2\)- \(4 \cdot (-3) = -12\)This gives the expression \(8x^2 - 12\).

Step 3: Multiply Second Monomial

Next, multiply \(x\) by each term in \((2x^2-3)\): - \(x \cdot 2x^2 = 2x^3\)- \(x \cdot (-3) = -3x\)This gives the expression \(2x^3 - 3x\).

Step 4: Combine Like Terms

Now, add the two expressions obtained from the previous steps: \(2x^3 + 8x^2 - 3x - 12\). There are no like terms to combine, so this is the final expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us efficiently handle terms in polynomials. Essentially, it allows you to multiply a single term across a set of terms within parentheses. In our exercise, the distributive property is used to simplify the multiplication of two polynomials:

• For the expression \((4+x)(2x^2-3)\), we distribute each term in \((4+x)\) to \((2x^2-3)\).
• This means multiplying \(4\) by each term inside the parentheses of the second polynomial and doing the same for \(x\).

Using the distributive property makes the complex look simple, because you break down tasks into manageable multiplication problems. By practicing this property in polynomial multiplication, you'll be able to handle more complex expressions with ease.

Monomials

When dealing with polynomials, it's important to understand the role of monomials, which are individual terms used in these expressions. A monomial is a product of a number and variable(s) raised to a power, with no plus or minus signs linking it to other variables or constants. In our example, we see monomials such as \(4\) and \(x\) in the first polynomial, and \(2x^2\) and \(-3\) in the second.

• Monomials play a crucial role in distributing and multiplying as each term must be treated independently.
• When multiplying, like in our example, you apply arithmetic to constants and laws of exponents to powers.

Understanding monomials simplifies the process of polynomial operations because they form the building blocks of polynomials, making complex expressions more manageable.

Combining Like Terms

Combining like terms is a critical skill in simplifying algebraic expressions, especially when dealing with multiple terms from polynomial multiplication. Like terms are terms that have the same variable raised to the same power. In our example exercise:

• After expanding the polynomial expressions, you assess if any terms are like terms, meaning they can be combined to simplify the expression.
• In the expression \((2x^3 + 8x^2 - 3x - 12)\), no terms have identical variable parts.

When like terms are present, you simply add or subtract their coefficients to reduce the expression efficiently. Regular practice in identifying and combining like terms aids in streamlining algebraic expressions, improving your overall problem-solving efficiency.