Applying The Rational Zero Theorem to Find The Zeros in a Polynomial Function.

You are in front of a polynomial, and you are supposed to find all the rational zeros for that polynomial. You are not allowed to use a graphing calculator. You know that you may apply the Rational Zero Theorem, but all the examples in the book skip a lot of steps and imply that you should know those steps, so they don't bother to write them anywhere in the textbook.

Well, here we come to the rescue! You will see the solution of three examples. They are just a few, but they show in every detail each one of the steps you should follow to accomplish the task. After you study them, you will find that is going to be much easier for you to tackle this type of problems; and something to add: You may go at your own pace! Important to suggest that every good student tries the solutions first by itself before getting them from the teacher, the book or in this case the webpage. You might try to take advantage of the MARKER TOOLS menu on the edge of the screen and work the problems with you finger or stylus on the screen. Besides you may scroll down and study all the formal definitions for different terms that apply or relate to the lesson.

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Lesson's Glossary

Binomial: Expression that has two (bi) terms.

Depressed polynomial: The resulting numbers of the synthetic division that correspond to the quotient of the division; these are the coefficients for the Depressed polynomial that is one degree less than the divisor polynomial.

Complex Conjugate Theorem: It states that if we have an imaginary root in a one variable polynomial with real coefficients, then we have another root at the complex conjugate of this root. So if we have a + bi, the other root is at a - bi.

Constant: It is a number or value that remains always the same. Never changes.

Descartes' rule of signs:
The rule states that ordering the terms of a single-variable polynomial  with real coefficients in descending order as to the exponents,  then we have that the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or less than it by a multiple of 2. This multiple of 2 refers to the fact that the number of imaginary roots is always even.  Multiple roots of the same value are counted separately. To find the number of negative roots we substitute the variable by its opposite (if variable is x then -x); then the number of negative roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients or less than it by a multiple of 2.

Difference of two square: a2 - b2 = (a +b)(a - b)

Exponent: It is a raised number representing the repeated multiplication of a given factor. Perfect square trinomial: A trinomial generated by the product of two equal binomials.

Expression: Any combination of numbers and operations without the = sign.

Factoring: The process to brake a polynomial down into the product of several factors.

Factors: All whole numbers that are multiplied together to yield another number.

Factored Form: Any polynomial that is written as the product of polynomials of lower degree that may be obtained from the original polynomial.

Like terms: Terms that have the same combination of variables to the same power as factors.

Polynomial: A algebraic statement with one or more terms. Word comes from “poly” which means many.

Polynomial (third degree): Geometric representation. A third degree polynomial may be represented as the volume of a rectangular prism for which length, width and depth are the linear factors of the polynomial.

Power: Exponent of a number or variable.

Rational number: Any number that may written as a fraction; including whole numbers (written as fractions with 1 as denominator) and decimals that truncate or repeat (may be expressed as fractions).

Rational root theorem: A theorem which states that in a polynomial the different combinations of the factors of the numerator and denominator of the quotient [constant term]/[leading coefficient] contains  all the possible rational  real zeros in the polynomial. In other words: If 0 = anxn + an-1xn-1 +...+ a1x + a0  then p/q in its simplest form is a rational root of this polynomial equation, where all coefficients an ...a0 are integers, and p must be a factor of a0 and q must be a factor of an .

Solution or root: The value that makes an equation a true statement, a root refers particularly to the value of x for which y = 0, this value is also the x-intercept of the graph.

Subtraction: Adding the opposite.

Synthetic division: A process of dividing a polynomial by a linear factor, using the coefficients and ignoring the variable and the exponents (they are relevant just as the position to place the coefficients). When dividing the sign of the divisor is reversed to avoid subtracting and to allow just to add.

Term: A form of grouping one or more numerical and/or variable factors by means of multiplication and division. Addition and subtraction symbols separate terms.

Variable: A letter used to represent a number. When the variable is part of an equation, it is possible to find the value for which it stands for by solving the equation. This is the solution (s) of the equation.

Variable expression: Mathematical phrase with at least one variable in it.

Zeros of a function: The solutions for the equation of the function when this equal to 0. The roots, also known as the x-intercepts.

Zero Product Property: For the product of any two real numbers; it means that at least one of them is zero, so the product be equal to zero. This property allow us to solve equations that are equal to zero; to be solved by factoring (linear factors)

 

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