Problem 53 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

Answer: \(a_0 = 12\)

Step 1: Determine the product h(x)

To find the product of \(f(x)\) and \(g(x)\), we will multiply the two functions. \(f(x) = 2x^3 + 3x - 3\) \(g(x) = 3x^2 - 2x - 4\) \(h(x) = f(x) * g(x)\) Distribute each term of \(f(x)\) with each term of \(g(x)\): \(h(x) = (2x^3)(3x^2) + (2x^3)(-2x) + (2x^3)(-4) + (3x)(3x^2) + (3x)(-2x) + (3x)(-4) + (-3)(3x^2) + (-3)(-2x) + (-3)(-4)\)

Step 2: Simplify h(x)

Now we need to combine like terms and simplify the expression for h(x): \(h(x) = 6x^5 - 4x^4 - 8x^3 + 9x^3 - 6x^2 - 12x - 9x^2 + 6x + 12\) Combine like terms: \(h(x) = 6x^5 - 4x^4 + x^3 - 15x^2 - 6x + 12\)

Step 3: Identify the constant term a0

The constant term of a polynomial is the term without any variable, which is the term \(a_0\) in the expression: \(h(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0\) Looking at the simplified version of \(h(x)\), we can identify the constant term as \(a_0\): \(h(x) = 6x^5 - 4x^4 + x^3 - 15x^2 - 6x + 12\) \(a_0 = 12\) The constant term \(a_0=12\) in the polynomial \(h(x)\). So the final answer is \(a_0 = 12\).

Key Concepts

Polynomial Multiplication

When working with polynomials, an essential skill to master is polynomial multiplication. This process involves multiplying two polynomial expressions together, resulting in a new polynomial. To correctly multiply polynomials, apply the distributive property. Each term in the first polynomial should be multiplied by every term in the second polynomial.

• Ensure you align each term properly as you distribute.
• Pay close attention to the exponents, as you will need to add them when multiplying variables.

For example, when multiplying the polynomials \(f(x) = 2x^3 + 3x - 3\) and \(g(x) = 3x^2 - 2x - 4\), you should handle it carefully:

• Multiply \(2x^3\) by each term in \(g(x)\).
• Do the same for \(3x\) and \(-3\).
• Combine all results into a single expression.

This step will lead to a long expression that can be quite complex, so clarity at each stage is vital.

Constant Term

The constant term in a polynomial is the term that does not contain any variables. It's an indispensable part of the polynomial, often regarded as \(a_0\) in standard polynomial notation. In the multiplication process of two polynomials,

• The constant term is found by summing the products of any constants from both expressions.
• It can also result from the cancellation of variable terms, simplifying down to a constant.

In our example, after multiplying \(f(x)\) and \(g(x)\), the constant term is found by identifying the term without an attached variable after the simplification:

For \(h(x) = 6x^5 - 4x^4 + x^3 - 15x^2 - 6x + 12\), the constant term is \(a_0 = 12\).

This specific term is vital for evaluations of the polynomial, as it provides the value when all other terms are zeroed out (when \(x = 0\)).

Combine Like Terms

Combining like terms is a crucial step in simplifying polynomials, particularly after multiplication. Like terms in a polynomial are terms that have the same variable part, with identical exponents, though the coefficients may differ.

• After distributing the terms in one polynomial across another, each resulting term should be grouped with others of the same degree.
• This grouping process allows for the simplification of the expression by combining coefficients.

From our multiplication example, the intermediate expression had terms that needed combining: \(h(x) = 6x^5 - 4x^4 - 8x^3 + 9x^3 - 6x^2 - 12x - 9x^2 + 6x + 12\).

Here, we needed to:

• Combine \(-8x^3 + 9x^3\) into \(x^3\).
• Combine \(-6x^2 - 9x^2\) into \(-15x^2\).
• Express \(-12x + 6x\) as \(-6x\).

This process reduces the polynomial to a simpler, easier-to-interpret form: \(6x^5 - 4x^4 + x^3 - 15x^2 - 6x + 12\). It ensures accuracy and clarity, refining the expression into one where each term's impact is easily understood.