Problem 34 - Algebra Form and Function

Question

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 3-2(x-5)^{2} $$

Step-by-Step Solution

Answer

Answer: The coefficients are \(a_{0} = -47, a_{1} = 20, a_{2} = -2\) and the degree is 2.

Step 1: Expand the given polynomial expression

First, we need to expand \((x-5)^{2}\). This can be done using the formula \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(a=x\) and \(b=5\). So, $$(x-5)^{2} = (x)^{2} - 2(x)(5) + (5)^{2} = x^2 - 10x + 25$$ Now, substitute this back into the given expression \(3-2(x-5)^{2}\): $$3 - 2(x^2 - 10x + 25)$$

Step 2: Distribute the -2

Now, distribute the -2 to each term in the parenthesis: $$3 - 2x^2 + 20x - 50$$

Step 3: Combine like terms

Combine the constants, 3 and -50: $$-47 - 2x^2 + 20x$$

Step 4: Write the polynomial in standard form

Rearrange the terms in decreasing powers of x and obtain: $$-2x^2 + 20x - 47$$

Step 5: Identify the coefficients and the degree of the polynomial

Now that we have the polynomial in standard form, we can easily identify the coefficients: \(a_{0} = -47, a_{1} = 20, a_{2} = -2\). Since the highest exponent of x is 2, the degree of the polynomial is 2.

Key Concepts

Standard Form

When working with polynomials, one of the primary goals is to express them in what is known as "standard form." This form helps make polynomials easier to read and work with. The standard form of a polynomial is expressed in descending order of the exponents on the variable by using the format:

• \(a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0}\)

Here, the term with the largest exponent comes first, followed by terms with smaller exponents, until you reach the constant term, which has no variable attached. In the exercise, after rearranging and simplifying the terms, the given polynomial was rewritten as:

• \(-2x^2 + 20x - 47\)

This is the standard form because the terms are ordered from the highest to the lowest power of \(x\). The term with \(x^2\) is first, followed by \(x\), then the constant.

Coefficients

Coefficients in polynomials are the numbers that multiply the variable terms. They are crucial as they define the effect each term has on the polynomial's graph and shape. The standard form of a polynomial makes identifying coefficients simple. In our polynomial expression:

• \(-2x^2 + 20x - 47\)

The coefficients are:

• \(a_2 = -2\)
• \(a_1 = 20\)
• \(a_0 = -47\)

These coefficients tell us how each term contributes to the overall polynomial. Importantly, they correspond directly to the terms:

• \(-2\) is the coefficient of \(x^2\)
• \(20\) is the coefficient of \(x\)
• \(-47\) is the constant term, or the coefficient of \(x^0\)

By identifying these numbers, you gain insight into how the polynomial behaves, particularly when graphed.

Degree of Polynomial

One important aspect of polynomials is understanding their "degree," which tells you the highest power (exponent) of the variable in the polynomial. This degree determines the general shape and behavior of the graph and also gives insight into the potential number of solutions a polynomial equation may have.In the polynomial we considered, \(-2x^2 + 20x - 47\), the largest exponent of \(x\) is \(2\), thus the degree is \(2\).

• This implies that the polynomial is quadratic, typically represented by a parabola shaped graph when plotted.
• A degree of \(2\) suggests that the polynomial can have up to 2 roots or solutions.

Degrees give us crucial insights into the polynomial's characteristics, and understanding it helps in various applications, including solving equations and modeling real-world phenomena.