Problem 29 - Algebra Form and Function

Question

(a) Find two different polynomials with zeros \(x=-1\) and \(x=5 / 2\). (b) Find a polynomial with zeros \(x=-1\) and \(x=\) \(5 / 2\) and leading coefficient \(4 .\)

Step-by-Step Solution

Answer

Question: Find two different polynomials with zeros at x = -1 and x = 5/2. Then, find a polynomial with zeros at x = -1 and x = 5/2, and a leading coefficient of 4. Answer: Two different polynomials with the given zeros are: 1. P1(x) = (x + 1)(x - 5/2) 2. P2(x) = 2(x + 1)(x - 5/2) The polynomial with the given zeros and a leading coefficient of 4 is: P(x) = 4x^2 - 6x - 10.

Step 1: Polynomial with given zeros

Since we are given that the zeros are \(x = -1\) and \(x = \frac{5}{2}\), we can express the polynomial as \((x + 1)(x - \frac{5}{2})\).

Step 2: Find first polynomial

The first polynomial can simply be the product of the factors from Step 1: \(P_1(x) = (x + 1)(x - \frac{5}{2})\).

Step 3: Find second polynomial

To find a different polynomial, we can multiply the expression from Step 1 by any non-zero constant. Let's multiply by \(2\): \(P_2(x) = 2(x + 1)(x - \frac{5}{2})\). So, two different polynomials with zeros at \(x = -1\) and \(x = \frac{5}{2}\) are: 1. \(P_1(x) = (x + 1)(x - \frac{5}{2})\) 2. \(P_2(x) = 2(x + 1)(x - \frac{5}{2})\) #Part (b)#

Step 1: Identify leading coefficient

Given that the leading coefficient is \(4\), this means that the highest-degree term in the polynomial will have a coefficient of \(4\).

Step 2: Find the constant to multiply

If we refer back to our polynomial expression with the given zeros, we have \((x + 1)(x - \frac{5}{2})\). Expanding this polynomial, we get: \(x^2 - \frac{3}{2}x - \frac{5}{2}\). The leading term is \(x^2\), which has a coefficient of \(1\). To achieve a leading coefficient of \(4\), we need to multiply the whole polynomial by the constant \(4\): $$ 4(x + 1)(x - \frac{5}{2}) $$

Step 3: Simplify polynomial

Lastly, to find our desired polynomial, we simplify the expression from Step 2: $$ 4(x^2 - \frac{3}{2}x - \frac{5}{2}) = 4x^2 - 6x - 10 $$ Thus, the polynomial with zeros at \(x = -1\) and \(x = \frac{5}{2}\), and a leading coefficient of \(4\) is: \(P(x) = 4x^2 - 6x - 10\).

Key Concepts

Leading Coefficient

The leading coefficient in a polynomial is an important aspect to understand, as it defines the coefficient of the term with the highest degree. In simpler terms, when you expand a polynomial expression, the leading coefficient is the number that comes in front of the variable raised to the highest power. For example, in the polynomial equation \(4x^2 - 6x - 10\), the leading coefficient is \(4\) because the term \(4x^2\) is the one with the highest degree, which is 2 here.

Understanding the leading coefficient is crucial because it affects the shape and the direction of a polynomial graph. In our exercise, knowing that the leading coefficient should be \(4\) helps us adjust the whole polynomial expression accordingly.

To find a polynomial with a specified leading coefficient, you often start with a basic polynomial that has the same zeros and multiply the whole expression by a constant to reach the desired leading coefficient.

Polynomial Expression

A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In other words, it's a combination of terms like \(ax^n\), where \(a\) is a coefficient, \(x\) is a variable, and \(n\) is a non-negative integer exponent.

For example, the polynomial expression \((x + 1)(x - \frac{5}{2})\) has factors that will give you a polynomial equation when expanded. Each factor represents a root or zero of the polynomial. This means that if we substitute these zeros back into the polynomial expression, the result would be zero. For this particular example, the zeros are \(x = -1\) and \(x = \frac{5}{2}\).

Breaking down polynomial expressions into their component factors helps when you are analyzing or solving polynomial equations. Factoring is especially useful because it simplifies the process of finding zeros and helps achieve specific polynomial structures for given conditions, such as having a specified leading coefficient.

Factorization

Factorization is the process of breaking down a larger expression into products of simpler expressions or factors. With polynomials, this typically involves identifying zeros and expressing the polynomial as a product of factors.

Consider the polynomial given in the exercise; it was expressed as \((x + 1)(x - \frac{5}{2})\) based on its zeros at \(x = -1\) and \(x = \frac{5}{2}\). This factorized form is beneficial because if you substitute either zero into the expression, it results in zero, confirming their validity.

Factorization simplifies complex polynomials and helps reveal the roots directly, making it easier to perform operations like multiplication or division involving these polynomials. Knowing how to convert a polynomial from expanded form into its factorized form, or vice versa, is a key skill in algebra that is widely used in both solving equations and understanding polynomial behaviors.