Can Quadratic Techniques Solve Polynomial Equations?

So far you know that you can solve polynomial equations using the Rational Zero Theorem, or the Quadratic Formula if the polynomial equation involves a second degree polynomial expression; but ...What if I have to solve a 4th degree equation? or a third degree equation?...What about an equation with fractional exponents?

This lesson presents in an elegant way how you may solve those cases, by applying what you know about solving quadratic equations. What you do is to reduce all cases to "second degree" equations. This is done by variable substitution. Additionally, you have odd exponents to the third power that may allow you to apply sum and difference of cubes. You have for several of the problems the option to first attempt the solution. For this you use the pen in the MARKER TOOLS menu. That is all!

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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.

Completing the Square: Method that finds the constant term in an incomplete perfect square trinomial of a second degree equation to solve it.

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.

Function: A relation of the type that has exactly one value in the domain (independent variable) matching a value in the range (dependent variable).

Function notation: A function written with the symbol f(x) instead of y. It is read as f of x.

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.

Quadratic equation: An equation of the form

ax2 + bx + c = 0 where a, b, and c are real numbers and a is different from zero.

Quadratic formula: If ax2 + bx + c = and a is different from zero then the quadratic formula is given in terms of a, b, and c.

Quadratic function: Any function in the form of

f(x) = ax2 + bx + c where a is different from zero. The graph is a parabola and the largest exponent is 2.

Square Root: If x2 = y, then x is the square root of y. Square root is the opposite of square.

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.

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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