Roots & Zeros of Polynomials I, Study Guides, Projects, Research of Algebra

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Roots & Zeros of

Polynomials I

How the roots, solutions,

zeros, x - intercepts and

factors of a polynomial

function are related.

Polynomials

A Polynomial Expression can be a monomial or a sum of monomials. The Polynomial Expressions that we are discussing today are in terms of one variable.

In a Polynomial Equation, two

polynomials are set equal to each other.

Since Factors are a Product...

…and the only way a product can equal zero is if one or more of the factors are zero… …then the only way the polynomial can equal zero is if one or more of the factors are zero.

Solving a Polynomial Equation The only way that x 2 +2 x - 15 can = 0 is if x = - 5 or x = 3 Rearrange the terms to have zero on one side: 2 2 x  2 x  15  x  2 x  15  0 Factor: ( x  5)( x  3)  0 Set each factor equal to zero and solve: ( 5) 0 and ( 3) 0 5 3 x x x x       

Zeros of a Polynomial Function A Polynomial Function is usually written in function notation or in terms of x and y.

f ( x )  x

2

 2 x  15 or y  x

2

 2 x  15

The Zeros of a Polynomial Function are the solutions to the equation you get when you set the polynomial equal to zero.

Zeros of a Polynomial Function The Zeros of a Polynomial Function ARE the Solutions to the Polynomial Equation when

the polynomial equals zero.

y  x 2  2 x  15

x - Intercepts of a Polynomial

The points where y = 0 are called the x - intercepts of the graph. The x - intercepts for our graph are the points... (-5, 0) and (3, 0)

x - Intercepts of a Polynomial

When the Factors of a Polynomial

Expression are set equal to zero, we get

the Solutions or Roots of the Polynomial

Equation.

The Solutions/Roots of the Polynomial Equation are the x - coordinates for the x-Intercepts of the Polynomial Graph!

Roots & Zeros of Polynomials II Finding the Roots/Zeros of Polynomials:

• The Fundamental Theorem of Algebra
• Descartes’ Rule of Signs
• The Complex Conjugate Theorem

Fundamental Theorem Of Algebra Every Polynomial Equation with a degree higher than zero has at least one root in the set of Complex Numbers. A Polynomial Equation of the form P(x) = 0 of degree ‘n’ with complex coefficients has exactly ‘n’ Roots in the set of Complex Numbers. COROLLARY:

Real/Imaginary Roots

Just because a polynomial has ‘n’ complex roots doesn’t mean that they are all Real! y  x 3  2 x 2  x  4 In this example, however, the degree is still n = 3 , but there is only one Real x - intercept or root at x = - 1, the other 2 roots must have imaginary components.

Descartes’ Rule of Signs

Arrange the terms of the polynomial P(x) in descending degree:

• The number of times the coefficients of the terms of P(x) change sign = the number of Positive Real Roots (or less by any even number)
• The number of times the coefficients of the terms of P(-x) change sign = the number of Negative Real Roots (or less by any even number) In the examples that follow, use Descartes’ Rule of Signs to predict the number of + and - Real Roots!

Find Roots/Zeros of a Polynomial If we cannot factor the polynomial, but know one of the roots, we can divide that factor into the polynomial. The resulting polynomial has a lower degree and might be easier to factor or solve with the quadratic formula. We can solve the resulting polynomial to get the other 2 roots: f ( x )  x 3  5 x 2  2 x  10 one root is x  5 x  5 x 3  5 x 2  2 x  10 x 3  5 x 2  2 x  10  2 x  10 0 x 2  2  (x - 5) is a factor x  2,  2

Complex Conjugates Theorem Roots/Zeros that are not Real are Complex with an Imaginary component. Complex roots with Imaginary components always exist in Conjugate Pairs. If a + bi ( b ≠ 0) is a zero of a polynomial function, then its Conjugate, a - bi , is also a zero of the function.