Problem 51 - Algebra Form and Function

Question

Evaluate the expressions in Problems \(51-54\) given that \(f(x)=2 x^{3}+3 x-3, \quad g(x)=3 x^{2}-2 x-4\) \(h(x)=f(x) g(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) $$ n $$

Step-by-Step Solution

Answer

Please provide the specific problem(s) from Problems 51-54 that you would like to evaluate based on the found polynomial form of h(x) = 6x^5 - 4x^4 + x^3 - 15x^2 - 6x + 12.

Step 1: Find \(h(x)\) as a product of \(f(x)\) and \(g(x)\)

To find the function \(h(x)\), we need to multiply the functions \(f(x)\) and \(g(x)\). So, we have: $$ h(x) = f(x)g(x) = (2x^3 + 3x - 3)(3x^2 - 2x - 4) $$

Step 2: Expand \(h(x)\)

To find the polynomial form of \(h(x)\), we should expand it by applying the distributive property (also known as the FOIL method – First, Outer, Inner, and Last): $$ h(x) = 2x^3(3x^2) + 2x^3(-2x) + 2x^3(-4) + 3x(3x^2) + 3x(-2x) + 3x(-4) -3(3x^2) - 3(-2x) - 3(-4) $$

Step 3: Simplify and rewrite the expanded expression

Now we need to simplify the terms obtained from the expansion and rewrite the expression in the form \(h(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0\): $$ h(x) = 6x^5 - 4x^4 - 8x^3 + 9x^3 - 6x^2 - 12x - 9x^2 + 6x + 12 $$

Step 4: Combine like terms

We'll combine the like terms and get the final expression for \(h(x)\): $$ h(x) = 6x^5 - 4x^4 + x^3 - 15x^2 - 6x + 12 $$ Now that we have found the polynomial form of \(h(x)\), we can evaluate the expressions for Problems 51-54. Please specify which problem(s) you would like to evaluate.

Key Concepts

Distributive Property

The distributive property is a cornerstone of polynomial multiplication. It's a rule that helps us multiply a single term by two or more terms inside a set of parentheses. When we talk about polynomial multiplication, the distributive property allows us to systematically break down the multiplication process.

Here's the principle: if you have an expression like \(a(b + c)\), you can distribute the multiplication over each addend inside the parentheses. Therefore, it can be rewritten as \(ab + ac\).

In the case of multiplying two polynomials, such as in our original exercise, this property is applied multiple times. For every term in the first polynomial, you distribute it across every term in the second polynomial. This is why understanding and correctly applying the distributive property is crucial when expanding polynomials.

• Break down terms using distribution.
• Consistently apply multiplication across terms.
• Result in each individual term which is subject for further simplification.

Like Terms

Like terms play a significant role when simplifying polynomial expressions. They are terms that share the same variables raised to the same powers. When simplifying, the coefficients (numbers in front of these terms) can be added or subtracted.

In our polynomial multiplication example, after expanding using the distributive property, we combined similar terms to simplify the polynomial. This means:

• Checking for terms with the same variable and degree.
• Adding or subtracting the coefficients of these terms.

For instance, in the expression \(9x^3 - 8x^3\), both terms have the variable \(x^3\). You can combine them to get \(1x^3\) or simply \(x^3\). This simplification process helps in rewriting the polynomial in its standard form, which is easier to handle.

Polynomial Expansion

Polynomial expansion involves using the distributive property to multiply each term in a polynomial by every term in another polynomial and then simplifying the result. It allows us to arrive at a single polynomial expression starting from the multiplication of two polynomial expressions.

The process of expansion can be broadly summarized as follows:

• Identify each term in the polynomials being multiplied (every term from the first should be multiplied with every term from the second).
• Apply the distributive property to conduct the multiplication between the terms.
• Simplify the expanded expression by combining like terms, resulting in a single, cleaner polynomial.

For example, expanding \( (2x^3 + 3x - 3)(3x^2 - 2x - 4) \) follows these steps, leading to a final polynomial form where facing a cleaner and organized expression for interpretation.