Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, find the product. \(\left(14 c^{2}+4 c\right)\left(2 c^{2}-3 c\right)\)

Step-by-Step Solution

Answer

The product is \(28c^4 - 34c^3 - 12c^2\).

Step 1: Apply the Distributive Property

To find the product of two binomials, use the distributive property. Distribute each term in the first binomial across the second binomial. Start with the first term in the first binomial, which is \(14c^2\). Distribute \(14c^2\) to \(2c^2 - 3c\):\[14c^2(2c^2 - 3c) = 14c^2 \cdot 2c^2 + 14c^2 \cdot (-3c)\].

Step 2: Simplify the First Distribution

Simplify each term from the distribution in Step 1. Multiply \(14c^2\) by \(2c^2\) to get \(28c^4\) and multiply \(14c^2\) by \(-3c\) to get \(-42c^3\). The expression becomes \(28c^4 - 42c^3\).

Step 3: Distribute the Second Term of the First Binomial

Now distribute the second term of the first binomial, \(4c\), across the second binomial. Perform \(4c(2c^2 - 3c)\):\[4c \cdot 2c^2 + 4c \cdot (-3c)\].

Step 4: Simplify the Second Distribution

Simplify the results from Step 3. Multiply \(4c\) by \(2c^2\) to get \(8c^3\), and \(4c\) by \(-3c\) to get \(-12c^2\). The expression becomes \(8c^3 - 12c^2\).

Step 5: Combine Like Terms

Add the results from Step 2 and Step 4 together: \(28c^4 - 42c^3 + 8c^3 - 12c^2\). Combine like terms to simplify: \(28c^4 + (-42c^3 + 8c^3) - 12c^2 = 28c^4 - 34c^3 - 12c^2\).

Step 6: Final Step: Write the Final Answer

The product of the binomials is \(28c^4 - 34c^3 - 12c^2\).

Key Concepts

Distributive Property

The distributive property is a key mathematical principle used to simplify expressions and solve equations. It involves distributing each term within a set of parentheses across another set of terms. This principle is essential in polynomial multiplication, allowing us to systematically cover all combinations of terms. Here is how it works:

• For the given problem: \((14c^2 + 4c)(2c^2 - 3c)\), apply the distributive property by multiplying each term of the first polynomial individually with every term of the second polynomial.
• The first term, \(14c^2\), is distributed across the terms \(2c^2\) and \(-3c\), resulting in two partial products: \(14c^2 \cdot 2c^2\) and \(14c^2 \cdot (-3c)\).

This technique ensures no term is left out and successfully transforms a product of two binomials into an expanded polynomial expression.

Binomials

Binomials are algebraic expressions that contain exactly two terms connected by a plus or minus sign, and they are a foundational building block in algebra. For example, the expressions \(14c^2 + 4c\) and \(2c^2 - 3c\) are both binomials. Understanding the structure of binomials is crucial when applying the distributive property for multiplication.
When multiplying binomials, every term in the first binomial must interact with every term in the second binomial. This is essential to ensure correct expansion, as seen in the steps for multiplying our example binomials. Here's a breakdown:

• Identify each term in both binomials. For \(14c^2 + 4c\): the terms are \(14c^2\) and \(4c\).
• For \(2c^2 - 3c\): the terms are \(2c^2\) and \(-3c\).

Once terms are identified, leverage the distributive property to carry out the multiplication process, ensuring each term multiplies with every term from the other binomial.

Combine Like Terms

After distributing and expanding polynomial expressions, the next essential step is to combine like terms. Like terms are terms that have the same variable part raised to the same power. Combining them simplifies the expression to its most compact form. This process is crucial for finding the most simplified answer after polynomial multiplication.
For instance, in our expanded expression \(28c^4 - 42c^3 + 8c^3 - 12c^2\), the like terms need to be combined:

• The terms \(-42c^3\) and \(+8c^3\) are like terms because they both contain the variable \(c\) raised to the power of three, so we combine them to \(-34c^3\).
• Keep the other terms as they have no like counterparts: \(28c^4\) and \(-12c^2\).

Therefore, combining like terms transforms the expression into \(28c^4 - 34c^3 - 12c^2\), which is the final simplified form of the product. This step is critical for ensuring clarity and accuracy in presenting algebraic solutions.