Polynomial Functions: Finding zeros, solutions, or roots.

How may I use synthetic division to determine that a linear binomial is a factor of a polynomial, or what is the use of the Descartes' Rule of Signs? How will the Complex Conjugate Theorem help me to find zeros? Given a number of zeros; may I get the polynomial function of least degree with integral coefficients?

This lesson has been rated very high by students when taught in the classroom. It shows each one of the steps of these laborious processes. It will dissipate any possible questions you may have! Why not trying to write you own solution after you study the first problems. You may use the pen on the MARKER TOOLS menu on the top of the screen. Very important is to take advantage of the vocabulary definitions given at the bottom of the page. As you move through the lesson you may find useful to study them or even better before you go through the lesson.

Lesson's Content

Lesson In PDF Format (no animations)


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.

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.

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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