Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (3 x-4)(2 x+9) $$

Step-by-Step Solution

Answer

The polynomial is \(6x^2 + 19x - 36\).

Step 1: Expand the Expression

To begin, we need to expand the expression \((3x - 4)(2x + 9)\) using the distributive property. This involves multiplying each term in the first binomial by each term in the second binomial. Therefore, we'll have:1. Multiply \(3x\) by \(2x\), resulting in \(6x^2\).2. Multiply \(3x\) by \(9\), resulting in \(27x\).3. Multiply \(-4\) by \(2x\), resulting in \(-8x\).4. Multiply \(-4\) by \(9\), resulting in \(-36\).

Step 2: Combine Like Terms

Now, we have all the individual terms from the expansion: \(6x^2\), \(27x\), \(-8x\), and \(-36\).We need to combine the like terms (terms with the same variable to the same power):- The \(x\) terms are \(27x\) and \(-8x\). Combine these to get:\(27x - 8x = 19x\).So, the polynomial becomes: \[6x^2 + 19x - 36\].

Key Concepts

Distributive Property

In mathematics, the distributive property is a crucial concept that helps in simplifying expressions and solving equations. It is a rule that allows you to multiply a term by a sum or difference within parentheses.
For example, in the expression \((3x - 4)(2x + 9)\), you apply the distributive property by multiplying each term in the first binomial by each term in the second binomial.
This looks like:

• Multiply \(3x\) by \(2x\) to get \(6x^2\).
• Multiply \(3x\) by \(9\) to get \(27x\).
• Multiply \(-4\) by \(2x\) to get \(-8x\).
• Multiply \(-4\) by \(9\) to get \(-36\).

By distributing each term across the binomials, you expand the expression into separate terms that can further be simplified or combined. This property is foundational in polynomial expansion and helps in organizing and solving algebraic expressions efficiently.

Like Terms

Like terms are terms that contain the same variables raised to the same power. Recognizing and combining like terms is essential for simplifying polynomials. In the polynomial \(6x^2 + 27x - 8x - 36\), the like terms are

• Terms involving \(x\): \(27x\) and \(-8x\).

You combine these like terms by adding or subtracting their coefficients. For instance:

• \(27x - 8x = 19x\)

By combining like terms, you simplify the polynomial, making it easier to use in further calculations or to interpret. This step is vital in ensuring your final expression is concise and accurate.

Binomial Multiplication

Binomial multiplication is a fundamental algebraic process that involves multiplying two binomials. Each binomial is an algebraic expression containing two terms. When multiplying binomials, as seen in the expression \((3x - 4)(2x + 9)\), you use an approach similar to the distributive property, known as the "FOIL" method.
FOIL stands for:

• First – Multiply the first terms of each binomial: \(3x \times 2x = 6x^2\)
• Outer – Multiply the outer terms: \(3x \times 9 = 27x\)
• Inner – Multiply the inner terms: \(-4 \times 2x = -8x\)
• Last – Multiply the last terms: \(-4 \times 9 = -36\)

This approach ensures that all combinations of terms from the binomials are multiplied, giving you the expanded form \(6x^2 + 27x - 8x - 36\) before combining like terms. Binomial multiplication is a core skill in algebra that lays the groundwork for more complex polynomial operations.