# Roots of Polynomial Equations

**Homework:
**12.2 #25, 37, 45, 49, 55

12.3 #13b, 15, 19, 35, 45, 55, 57

12.4 #2, 19, 25, 28

**Review:** An equation is a mathematical statement
that two expressions are

equivalent. A root or solution of an equation is a value (values) such that,

when substituted for the variable (variables) in the equation, the

mathematical statement is true. To solve an equation means to find its

solutions (roots). Equations may have no solutions, a finite number of

solutions , or an infinite number of solutions. Solutions may be real or

complex, single or multiple. We can rewrite an equation in an equivalent

form (i.e., where the solution set is the same) as Expression = 0 .

polynomial equations , roots. The solutions to a polynomial
equation,

a_{n} x^{n} + ...+ a_{1}x + a_{0} = 0, are called the roots of the polynomial. Some

polynomial equations do not have any roots among the real numbers , e.g.,

x^{2} + 1 = 0 . Some have repeated or multiple roots, e.g., (x - 1)^{2} = 0 .

Some have distinct roots, e.g., x^{2} - 1 = 0 . And some have all of the

above, e.g., x^{2} ( + 1) x^{2} ( - 1)(x - 1)^{2} = 0 .

division algorithm for polynomials. The division algorithm
for numbers states that

the dividend is equal to the product of the divisor and the quotient added

to the remainder, where the remainder is less than the divisor

(Dividend = Divisor -Quotient + Remainder ). Similarly, the division

algorithm for polynomials states that f (x) = d(x) · q(x)
+ R(x), where the

degree of the remainder, R(x), is less than the degree of the divisor

d(x). Let f (x) = x^{3} + 7x^{2} - x +11 and d(x) = x - 5. Find a polynomial

q(x) and a constant remainder, R(x), such that f (x) = d(x) ·
q(x) + R(x) or

(equivalently)

divide polynomials using long and synthetic division . We
can use synthetic

division to divide polynomials when our factor is linear of the form

x - r . Find the quotients and remainders for:

(1) 9x^{4} + x^{3} -12x + 21 divided by x + 4

(2) 3r^{4} +16r^{3} - 5r +19 divided by 3r - 2

remainder theorem. The division algorithm for polynomials
states that for

linear divisors , x - r, f (x) = (x - r) · q(x) + R(x)
where the degree of the

remainder, R(x), is zero . The Remainder Theorem states when a

polynomial f (x) is divided by x - r , the remainder is f(r). [Substitute r

for x.] What is the remainder when f (x) = x^{4} - 3x^{3} + 7x^{2} - x + 5 is

divided by x - 3? Without substituting, what is f (3)?

(linear) factor theorem. The (Linear) Factor Theorem
states: Let f (x) be a

polynomial. If f (r) = 0, then x - r is a factor of f (x) . Conversely, if x - r

is a factor of f (x) , then f (r) = 0. We know even more. If (x - r_{k} ) is a

factor of f (x) , then:

• f (x) is divisible by (x - r_{k} )

• r_{k} is a zero of f (x) , i.e., f (r_{k} ) = 0

• r_{k} is a root of

• r_{k} is a x-intercept of the graph of f (x)

The (Linear) Factor theorem can be used to obtain a
reduced equation and,

thus, to find the remaining roots. We divide f(x) by x - r to get f (x) = (x -
r) · q(x), where

q(x) is the quotient function. Since q is one degree less than f, hopefully it
is easier to solve

q(x) = 0 than the original equation

f (x) = 0. Q(x) is called a reduced equation of f. If 3 and -2/3 are

roots of equation 3x^{4} + 5x^{3} - 49x^{2} +11x + 30 = 0, find the other roots.

fundamental theorem of algebra . Does every polynomial
equation of

degree at least 1 have a root ? Or, to pose the question differently, is it

possible to write a polynomial equation that has no solutions? If we allow

only real solutions, then we can easily write an equation with no real roots:

x^{2} = -1. However, if we allow complex roots, then it turns out that every

polynomial equation has at least one root. This is known as the

Fundamental Theorem of Algebra. This theorem applies not only to

polynomials with rational number coefficients

but also to polynomials with real number coefficients

and
with complex number coefficients (x^{2} - 2ix + [3+ i] = 0).

linear factors theorem. According to the Fundamental
Theorem of Algebra, the

equation f (x) = a_{n} x^{n} + a_{n-1}x^{n-1} + ...+ a_{1}x + a_{0} = 0 has at least one

(possibly complex) root. Let's call this root r_{1}. By the Factor Theorem,

x - r_{1}, is a factor of f (x) , and we can write f (x) = (x - r_{1})Q_{1}(x) , where

Q_{1}(x) is a polynomial of degree n -1 and leading coefficient a_{n} . If the

degree of Q_{1}(x) is zero, i.e., it is a constant, then were done and

Q_{1}(x) = a_{n} . If not then we can write Q_{1}(x) = (x - r_{2})Q_{2}(x) for a Q_{2}(x) of

degree n - 2 and leading coefficient a _{n}. Continuing this process, we can

write f (x) = a_{n} (x - r_{1})(x - r_{2})...(x - r_{n} ) . Therefore, a polynomial
equation

of degree n ≥ 1 has exactly n roots (some possibly complex), where some

roots may be repeated (multiple) roots. We can factor

6x^{2} + x -12 as (3x - 4)(2x + 3). We can find the roots of

6x^{2} + x -12 = 0 by setting each factor equal to zero. Therefore, the roots are

x = 4/3, x = -3/2. Finally, we write
We
can

also use this theorem to find polynomial functions with given roots. Find

a quartic polynomial with roots

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