Question

In Problems 50 and 51 , if \(f(x)=x^{2}+2 x-7\) and \(g(x)=3 x-4\) find: 51\. \((f \cdot g)(x)\)

Short answer

(f · g)(x) = 3x^3 + 2x^2 - 29x + 28

Step-by-step solution

Write Down the Functions

First, note the given functions: \[ f(x) = x^2 + 2x - 7 \] and \[ g(x) = 3x - 4 \]

Define the Product of the Functions

The product of the functions \(f \cdot g\)(x) is defined as: \[ (f \cdot g)(x) = f(x) \cdot g(x) \]

Multiply the Functions

To find \(f \cdot g\)(x), multiply \( f(x) \) by \( g(x) \): \[ (f \cdot g)(x) = (x^2 + 2x - 7) \cdot (3x - 4) \]Distribute each term in \((3x - 4)\) by each term in \((x^2 + 2x - 7)\).

Distribute and Combine Like Terms

Perform the multiplication: \[ (x^2 + 2x - 7) \cdot (3x) = 3x^3 + 6x^2 - 21x \] and \[ (x^2 + 2x - 7) \cdot (-4) = -4x^2 - 8x + 28 \] Combine these results: \[ (f \cdot g)(x) = 3x^3 + 6x^2 - 21x - 4x^2 - 8x + 28 \] Combine like terms: \[ (f \cdot g)(x) = 3x^3 + 2x^2 - 29x + 28 \]

Key concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers and coefficients. They often appear in the form:

• a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

where each a_i is a coefficient and x is the variable. In the given problem, we have two polynomial functions: f(x) = x² + 2x - 7 and g(x) = 3x - 4.Notice that f(x) is a second-degree polynomial (highest power is x²) and g(x) is a first-degree polynomial (highest power is x).These forms are important because they dictate how the functions behave and how we can manipulate them, like in the multiplication we are doing here.

Distributive Property

The distributive property is essential when multiplying polynomials. It states that c(a + b) = ca + cb.This means we need to multiply each term in the first polynomial by every term in the second polynomial. Applying this to our problem, we multiply f(x) = x² + 2x - 7 by g(x) = 3x - 4in stages:

• Multiply each term of (3x - 4) by each term of (x² + 2x - 7).

This looks like:

• (x² + 2x - 7) * 3x = 3x³ + 6x² - 21x
• (x² + 2x - 7) * (-4) = -4x² - 8x + 28

This step-by-step multiplication follows the distributive property, ensuring all terms are correctly multiplied.

Combining Like Terms

After applying the distributive property, we get a series of terms that need to be combined. Combining like terms means adding or subtracting terms with the same power of x. In our schedule, the results from (x² + 2x - 7) * 3x = 3x³ + 6x² - 21xand (x² + 2x - 7) * (-4) = -4x² - 8x + 28must be added together: Combining like terms:

• The x³ terms: 3x³
• The x² terms: 6x² - 4x² = 2x²
• The x terms: -21x - 8x = -29x
• The constant terms: 28

So,Combining all, we get: 3x³ + 2x² - 29x + 28.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. In our exercise, we work with two algebraic expressions: f(x) = x² + 2x - 7 and g(x) = 3x - 4.Multiplying these expressions as we did involves understanding how to handle the variables, coefficients, and terms. Understanding the structure of algebraic expressions is key to solving multiplication problems. Each step, such as applying the distributive property and combining like terms, relies on recognizing how algebraic structures interact. Working through exercises step by step improves proficiency in manipulating these expressions accurately.