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Chapter 12: Problem 28

Find two different polynomials of degree 3 with zeros \(1,2,\) and \(3 .\)

Short Answer

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Question: Find two different polynomials of degree 3, which have the same zeros: 1, 2, and 3. Answer: The two different polynomials of degree 3 with zeros 1, 2, and 3 are: 1. \(f_1(x) = x^3 - 6x^2 + 11x - 6\) 2. \(f_2(x) = 2x^3 - 12x^2 + 22x - 12\)

Step by step solution

01

Write a general polynomial with given zeros

Let's write a general polynomial of degree 3 with zeros \(1, 2,\) and \(3\): \(f(x) = k(x-1)(x-2)(x-3)\)
02

Find the first polynomial

We need to find two different polynomials, so we can choose different values for the constant \(k\). For the first polynomial, let's choose \(k=1\): \(f_1(x) = (x-1)(x-2)(x-3)\) Now, expand the polynomial: \(f_1(x) = (x-1)[(x-2)(x-3)] = (x-1)(x^2-5x+6) = x^3 - 6x^2 + 11x - 6\)
03

Find the second polynomial

For the second polynomial, let's choose a different value for the constant \(k\), say \(k=2\): \(f_2(x) = 2(x-1)(x-2)(x-3)\) Now, expand the polynomial: \(f_2(x) = 2(x-1)[(x-2)(x-3)] = 2(x-1)(x^2-5x+6) = 2x^3 - 12x^2 + 22x - 12\) So, the two different polynomials of degree 3 with zeros \(1, 2,\) and \(3\) are: 1. \(f_1(x) = x^3 - 6x^2 + 11x - 6\) 2. \(f_2(x) = 2x^3 - 12x^2 + 22x - 12\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degree of Polynomial
The degree of a polynomial is an essential concept in algebra that provides a lot of information about the polynomial. Simply put, it is the highest power of the variable in the polynomial expression. For example, in the polynomial \(x^3 - 6x^2 + 11x - 6\), the highest power of \(x\) is 3, making it a polynomial of degree 3.

Understanding the degree of a polynomial is crucial because it tells us several things about the polynomial:
  • The number of zeros or roots it can have (considering possible complex numbers as well).
  • The basic shape of its graph, especially the end behavior.
  • How many times the polynomial will intersect the x-axis if all roots are real and distinct.
Recognizing these features helps in sketching graphs, predicting behavior, and solving polynomial equations effectively.
Zeros of Polynomial
Zeros, or roots, of a polynomial are the x-values for which the polynomial equals zero. For example, in the polynomial \((x-1)(x-2)(x-3)\), the zeros are \(x = 1, 2,\) and \(3\). These represent points where the graph of the polynomial will intersect the x-axis.

Finding zeros is valuable for solving equations because it allows you to determine where the polynomial changes sign, which is useful in various applications such as optimization problems.
  • If the polynomial is factored, like \((x-1)(x-2)(x-3)\), it鈥檚 easy to spot the zeros just by setting each factor to zero.
  • If not factored, other methods like synthetic division, the quadratic formula (for degree 2), or computational methods might be needed.
Zeros also relate directly to the concept of factors - if \(r\) is a zero of a polynomial \(f(x)\), then \((x-r)\) is a factor of \(f(x)\). Knowing this helps in polynomial division and further expansions.
Polynomial Expansion
Polynomial expansion involves expanding a factored polynomial expression into a standard polynomial form, where powers of \(x\) are written explicitly. Starting with a factored polynomial, such as \((x-1)(x-2)(x-3)\), we practice multiplying out the terms to get a standard form like \(x^3 - 6x^2 + 11x - 6\).

Here鈥檚 why polynomial expansion is useful:
  • It provides a clearer view of the coefficients, which are important for graphing and understanding the behavior of the polynomial.
  • Helps in performing operations like addition or subtraction on polynomials, since like terms are easily combined.
  • Necessary for solving equations; sometimes, expanded form makes it easier to apply other algebraic techniques.
Expanding the polynomial lets us visually interpret and manipulate the expression better in various mathematical contexts, from calculus to more complex algebraic solutions.

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Most popular questions from this chapter

Give the leading coefficient. $$ 1-6 r^{2}+40 r-\frac{1}{2} r^{3}+16 r $$

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

Consider the polynomial \(p(x)=(x-k)^{n},\) where \(k\) is a constant and \(n\) is a positive integer. (a) If \(n\) is even explain why the graph of \(p(x)\) is never below the \(x\) -axis. (b) If \(n\) is odd explain why the graph of \(p(x)\) is below the \(x\) -axis for \(xk\).

Give all the solutions of the equations. $$ (u+3)^{3}=-(u+3)^{3} $$

Without solving the equation, decide how many solutions it has. $$ \left(x^{2}+2 x\right)(x-3)=0 $$
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