Module 8

Complex Conjugate Theorem

Complex (a+bi) zeros always come in conjugate pairs. If a+bi is a complex zero, so is a-bi

It could have a zero, but not enough information

Given a<b<c<d and f(a)>0, f(b)>0, f(c)<0, f(d)>0, what conclusion can be made about [a,b]

It could have a zero, but not enough information

Given a<b<c<d and f(a)>0, f(b)>0, f(c)<0, f(d)>0, what conclusion can be made about [a,d]

Yes, there is at least one zero

Given a<b<c<d and f(a)>0, f(b)>0, f(c)<0, f(d)>0, what conclusion can be made about [b,c]

Yes, there is at least one zero`

Given a<b<c<d and f(a)>0, f(b)>0, f(c)<0, f(d)>0, what conclusion can be made about [c,d]

0(2), 2(1), √5(1), −√5(1) touch, cross, cross, cross

Given f(x)=−3x²(x−2)(x²-5)(x²+1)³, find the real zeros and their multiplicity, along with whether they touch or cross the x-axis

P(x)=−3x¹¹ ↑↓

Given f(x)=−3x²(x−2)(x²-5)(x²+1)³, state the power function and end behavior

a) 2(1), -3(2), ½(4) b) P(x)=5x⁷

Given the function *f(x)=5(x-2)(x+3)²(x-½)⁴*, state the a) zeros and their multiplicity and b) the power function

a repeated factor or multiple zero of f

If a zero (r) or a factor (x-r) occurs more than 1 time then r is called a . . .

a zero, root, or solution

If f is a polynomial function and r is a real number for which f(r)=0, then what is r called?

r is an x-intercept and r is a factor

If r is a real zero of f, then . . .

The graph TOUCHES the x-axis at that zero

If r is a zero of even multiplicity, then . . .

The graph CROSSES the x-axis at that zero

If r is a zero of odd multiplicity then . . .

the end behavior is ↓↓ (ends point down)

If the power function of a polynomial is aₙ<0 and n is even then the end behavior is?

the end behavior is ↑↓ (ends point up and down)

If the power function of a polynomial is aₙ<0 and n is odd

the end behavior is ↑↑ (ends point up)

If the power function of a polynomial is aₙ>0 and n is even then the end behavior is?

the end behavior is ↓↑ (ends point down and up)

If the power function of a polynomial is aₙ>0 and n is odd then the end behavior is?

n-1

In a polynomial function f of n degrees, what is the maximum number of turning points this function can have?

f(1) = 1-1-1 = -1 f(2) = 32-8-1 = 23 f(1)<0 and f(2)>0, I have at least 1 zero between them

Show that f(x) = x⁵-x³-1 has at least one zero in [1,2]

Yes, no degree

State whether the given function is a polynomial or not. If yes, state the degree. If no, state why. F(x)=0

Yes, the degree is 0 (think of this like 8x⁰)

State whether the given function is a polynomial or not. If yes, state the degree. If no, state why. G(x)=8

Yes; the degree is 4

State whether the given function is a polynomial or not. If yes, state the degree. If no, state why. f(x)=2−3x⁴

No, b/c there is a variable in the square root

State whether the given function is a polynomial or not. If yes, state the degree. If no, state why. g(x)=√x

No, b/c no variables in the denominator

State whether the given function is a polynomial or not. If yes, state the degree. If no, state why. h(x)= (x²−2)/(x³-1)

1) Use the degree of the polynomial to determine the maximum number of zeros (real and imaginary) 2) Use Descartes' Rule of Signs to determine the possible number of positive real zeros and negative real zeros (PNI chart) 3) If the polynomial has integer coefficients, use the Rational Zero Theoream to identify the rational zeros that can potentially be zero 4) Use Synthetic Division to test each potential rational zeros 5) each time a zero (thus a factor) is found, repeat step 4 on the depressed polynomial 6) Check possible factors to see if there are multiple of that zero 7) In attempting to find the zeros, remember to use (if possible) the factoring techniques that you already should know (special products, factoring by grouping, and so on) 8) Once the depressed polynomial is a quadratic, you can always use the quad formula of factoring to find remaining zero(s)

Steps to finding the real zeros of a polynomial

f(x)=0

The real zeros of a polynomial function f are the real solutions, if any, of the equation. . .

Remainder Theorem

This theorem states that if polynomial P(x) is divided by x-a, then the remainder r is P(a)

Factor Theorem

This theorem states that if x-c is a factor of f, then f(c)=0

Positive: 2 or 0 Negative 1

Use Descartes' Rule of Signs to determine the possible number of positive and negative real zeros: f(x) = -3x⁷+4x⁴+3x²-2x-1 f(-x) = 3x⁷+4x⁴+3x²+2x-1

Positive: 3 or 1 Negative: 3 or 1

Use Descartes' Rule of Signs to determine the possible number of positive and negative real zeros: f(x) = 3x⁶-4x⁴+3x³+2x²-x-3 f(-x) = 3x⁶-4x⁴-3x³+2x²+x-3

(p = ±(1, 2, 3, 6)) (q = ±(1, 2)) p/q = ± (1, 2, 3, 6, ½, 3/2) (Rational zero theorem refers to p/q)

Use the Rational Zero Theorem to determine the possible rational zeros for each: f(x) = 2x³+11x²-7x-6

It determines the possible number of positive and negative rational zeros

What does Descartes' Rule of Signs do?

It determines the possible values of rational zeros

What does the Rational Zero Theorem do?

To determine whether that domain is above or below the x-axis

What is sign-analysis used for when graphinga polynomial?

(−∞, ∞)

What is the domain of *all* polynomial functions?

x= (-b±√b²-4ac)/2a

What is the quadratic formula?

the exponent around the (x-r) factor

Where does multiplicity come from?

It is the first term in a polynomial

Where is the power function in a polynomial?

zero function

a polynomial function with *no* degree

constant function

a polynomial function with a degree of 0

linear function

a polynomial function with a degree of 1

quadratic function

a polynomial function with a degree of 2

cubic function

a polynomial function with a degree of 3

quartic function

a polynomial function with a degree of 4

multiplicity

how many times does that factor happen in a polynomial

degree

A polynomial function cannot have more real zeros than its _____

Rational Zero Theorem

Let f be a polynomial function of degree 1 or higher of the form f(x) = aₙxⁿ+aₙ₋₁xⁿ⁻¹+. . .+a₁x+a₀ where aₙ≠0 and a₀≠0 and where each coefficient is an integer. If p/q in lowest terms, is a rational zero of f, the p must be a factor of a₀ and q must be a factor of aₙ

Descartes Rule of Signs

Let f be a polynomial function: -The number of positive real zeros of f either equals the number of variation in sign of the non-real coefficients of f(x) or else equals that number less an even integer (2) -The number of negative real zeros of f either equals the number of variations in sign of the non-real coefficients of f(-x) or else equals that number less and even integer (2)

Intermediate Value Theorem

Let f denote a polynomial function. If a<b (x-coordinates) and if f(a) and f(b) (y-coordinates) have opposite signs, then there is at least one zero of f between a and b

P(x) = axⁿ (where a is a real number and does not equal 0, and n is a positive integer)

Power function of degree n is in the form of. . .?