Fundamental Theorem of Algebra - Washington-Liberty...3.3 - Zeroes of Polynomials Fundamental Theorem of Algebra-Every polynomial of degree n will have n zeroes (real and complex/imaginary)

November 3, 2017

3.3 - Zeroes of PolynomialsFundamental Theorem of Algebra

-Every polynomial of degree n will have n zeroes (real and complex/imaginary)

Linear Factorization Theorem

Every polynomial p(x) with degree n can be written as product of linear factors where c are complex numbers:

p(x)=a(x-c1)(x-c2)....(x-cn)

The real numbers are a subset of the complex (when b = 0)

complex = a + bi

Example 1: Write p(x) = x4 - 3x2 - 4 as a product of linear factors and list all the zeroes.

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Example 2: Write p(x) = 5x3 + 2x2 - 10x - 4 as a product of linear factors and list all the zeroes.

Factor by group tip:

It works if (first)(fourth) = (second)(third)

Multiplicity of Zeroes:

Multiplicity is how many times a zero is repeated. You may have an even or odd multiplicity

(x-3)2(x+3)4(x-1)(x+2)5

Example 3: State the zeroes and their multiplicity:

a)

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b) (x2+10x+25)(x2+x-20)(x-4)(x-5)

We saw in the last notes that zeroes that are complex (or use square roots) always come in conjugate pairs.

That is if -8i is a zero, then ____ is also one.

If 7+2i is a zero, then ______ is also one.

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How to multiply complex/radical roots the easier way to produce the original polynomial:

Remember (a-b)(a+b) = a2 - b2

Example 4:

a) zeroes are 4i and -4i

b) zeroes are ⎷3 and -⎷3

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c) zeroes are 5+i and 5-i

d) zeroes are 3+2⎷5 and 3-2⎷5

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Example 5: Write the equation of a degree 4 polynomial when it is known that 2 is the only real zero and -3i is also a zero.

Example 6: Write the equation of a degree 3 polynomial knowing that it has zeros of 1 and (1+i⎷2 )

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Note: If a problem doesn't ask for linear factors, then it is okay to leave (x2+4) as a factor instead of writing (x-2i)(x+2i).

They call (x2+4) an irreducible factor since it produces no real zeroes.

You will often have to do this if a problem asks for real coefficients/numbers only.

Intermediate Value Theorem (IVT)

Given P is a polynomial with real coefficients, if P(a) and P(b) have opposite signs then there is at least one value c between a and b such that P(c) = 0

In other words if you switch from positive y values to negative y-values (or vice versa), then there is a place where y = 0

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Example 7: Use the Intermediate Value Theorem to show P(x) = x3 - 9x + 6 has at least one zero in the given interval:

a) [0, 1] b) [-4, -3]

If P(a) and P(b) are both positive, does that mean there are no zeroes in the interval [a, b]?

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We've seen two theorems so far:

1. The Fundamental Theorem tells us that the zeroes exist.

2. IVT tells how to check intervals for a zero.

The next theorem will help give us a list of possible rational zeroes for a polynomial

Rational Zeroes TheoremGiven a polynomial with integer coefficients, and (p/q) is a rational number in lowest terms, the rational zeroes (if they exist) must be in the form of (p/q) where p is a factor of the constant term, and q is a factor of the leading coefficient.

Keep in mind this will not give complex or radical zeroes!

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Example 8: List all the possible rational zeroes for 3x3+7x2-4

Let's use a calculator to find the zeroes of the y = 3x3+7x2-4

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Example 9: List all the possible rational zeroes for 12x2-5x-2

Steps for finding the real zeroes of a polynomial function

Step 1: Use the degree of the polynomial to determine maximum number of zeroes

Step 2: Use the Remainder Theorem, synthetic division, and/or long division to test potential zeroes. Each time a zero is found continue to do step 2 on the new equation.

Remember that you can still use factoring techniques!

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Example 10: Write the polynomial as product of linear factorsa) 4x4-15x3+9x2+16x-12

b) x4+2x3-5x2-4x+6

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c) 3x4-5x3+14x2-20x+8