Problem 27 - Algebra 2

Question

Write each expression in quadratic form, if possible. $$ 7 b^{5}-4 b^{3}+2 b $$

Step-by-Step Solution

Answer

The expression cannot be written in quadratic form due to the powers involved.

Step 1: Identify the general form of a quadratic expression

A quadratic expression generally has the form \(ax^2 + bx + c\). Any expression in which the highest power of a variable is 2 qualifies as a quadratic expression.

Step 2: Examine the expression and identify variable powers

The given expression is \(7b^{5} - 4b^{3} + 2b\). Note the variable powers: \(b^5, b^3,\) and \(b^1\). A quadratic expression can have only powers of \(2, 1,\) and a constant term with power of 0.

Step 3: Rearrange and check if quadratic form is possible

In its current form, with powers \(b^5\) and \(b^3\), it does not fit the quadratic form since the highest power is not \(2\). Therefore, it cannot be written in quadratic form.

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases that consist of variables raised to whole number powers and coefficients. Each part of the expression, separated by addition or subtraction, is called a term. For instance, in the expression \(7b^{5} - 4b^{3} + 2b\), there are three terms: \(7b^{5}\), \(-4b^{3}\), and \(+ 2b\).
These expressions can come in various forms and the degree of a polynomial is determined by the highest power of the variable present. In our given expression, the highest power is \(5\), making it a polynomial of degree 5.

• The first term is \(7b^{5}\) where \(7\) is the coefficient and \(b^{5}\) indicates that the variable \(b\) is raised to the fifth power.
• The second term is \(-4b^{3}\) with \(-4\) as the coefficient, and \(b^{3}\) denotes that \(b\) is raised to the third power.
• The last term is simply \(2b\) where \(2\) is the coefficient and \(b\) is raised to the first power.

Understanding polynomial expressions and their structure helps to determine if they can be rewritten in another form, like the quadratic form.

Variable Powers

Variable powers in a polynomial tell us the degree of the variable in that particular term. Each term of a polynomial expression will have a variable, typically denoted by a letter like \(b\), raised to some power, which is a non-negative integer.
In the expression \(7b^{5} - 4b^{3} + 2b\), the powers are \(5\), \(3\), and \(1\). These describe how many times the variable is multiplied by itself:

• \(b^{5}\): The variable \(b\) is used as a factor five times, which means it is multiplied by itself five times.
• \(b^{3}\): Here, \(b\) is a factor three times.
• \(b^{1}\): This is simply \(b\) as a factor, meaning the variable appears just once.

For a polynomial to be a quadratic expression, the exponents (or powers) used must include at most \(2, 1,\) and a constant, which implies \(b^0 = 1\). Our original polynomial does not have these qualities because its highest power is \(5\), well beyond a simple quadratic.

Quadratic Form

The quadratic form is a specific expression where the variable's highest power is \(2\). It takes the general structure of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Quadratic forms are prevalent because they often describe parabolas in graphs, have solutions through methods like factoring, and are foundations for numerous algebraic processes.
Quadratic expressions are essential for modeling various real-world scenarios like physics problems, area calculations, and finance. Unfortunately, in the exercise given, the expression \(7b^{5} - 4b^{3} + 2b\) cannot be written in this quadratic form because its highest power is a \(5\), not a \(2\). To be a proper quadratic form, the expression needs terms with powers of only \(2, 1,\) and a constant \(0\), none of which our current expression can accurately present.
Understanding when a polynomial can be rearranged into a quadratic form or why it cannot, like in this instance, is crucial for solving a wide range of algebraic problems effectively.