Presentation on theme: "HW: Pg. 175 #7-16. Polynomial Functions- ◦ Let n be a nonnegative integer and let a 0, a 1, a 2,…, a n-1, a n be real numbers with a n ≠0. The functions."— Presentation transcript:
HW: Pg. 175 #7-16
Polynomial Functions- ◦ Let n be a nonnegative integer and let a 0, a 1, a 2,…, a n-1, a n be real numbers with a n ≠0. The functions given by f(x)=a n x n + a n-1 x n-1 +…+a 2 x 2 +a 1 x+a 0 Is a polynomial function of degree n. The leading coefficient is a n. f(x)=0 is a polynomial function. *it has no degree or leading coefficient.
NameFormDegree Zero FunctionF(x) = 0Undefined Constant FunctionF(x)=a (a≠0)0 Linear FunctionF(x)=ax+b (a≠0)1 Quadratic FunctionF(x)=ax 2 +bx+c (a≠0)2
F(x) = ax+b Slope-Intercept form of a line: Find an equation for the linear function f such that f(-2) = 5 and f(3) = 6
The average rate of change of a function y=f(x) between x=a and x=b, a≠b, is [F(b)-F(a)]/[b-a]
Weehawken High School bought a $50,000 building and for tax purposes are depreciating it $2000 per year over a 25-yr period using straight-line depreciation. 1. What is the rate of change of the value of the building? 2. Write an equation for the value v(t) of the building as a linear function of the time t since the building was placed in services. 3. Evaluate v(0) and v(16) 4. Solve v(t)=39,000
Point of ViewCharacterization VerbalPolynomial of degree 1 AlgebraicF(x)=mx+b (m≠0) GraphicalSlant line with slope m and y- intercept b AnalyticalFunction with constant nonzero rate of change m, f is increasing if m>0, decreasing if m<0
Sketch how to transform f(x)=x 2 into: G(x)=-(1/2)x 2 +3 H(x)=3(x+2) 2 -1 If g(x) and h(x) and in the form f(x)=ax 2 +bx+c, what do you notice about g(x) and h(x) when a is a certain value (negative or positive)?
f(x)=ax 2 +bx+c We want to find the axis of symmetry, which is x=-b/(2a). Then: The graph of f is a parabola with vertex (x,y), where x=-b/(2a). If a>0, the parabola opens upward, and if a<0, it opens downward.
Now we can use this method to find both remainders and quotients for division by x-k, called synthetic division. (2x 3 -3x 2 -5x-12)/(x-3) K becomes zero of divisor
3 | _____________ * Since the leading coefficient of the dividend must be the leading coefficient, copy the first “2” into the first quotient position. * Multiply the zero of the divisor (3) by the most recent coefficient of the quotient (2). Write the product above the line under the next term (-3). * Add the next coefficient of the dividend to the product just found and record sum below the line in the same column. * Repeat the “multiply” and “add” steps until the last row is completed.
Suppose f is a polynomial function of degree n1 of the form f(x)=a n x n +…+a 0 with every coefficient an integer. If x=p/q is a rational zero of f, where p and q have no common integer factors other than 1, then ◦ P is an integer factor of the constant coefficient a 0, and ◦ Q is an integer factor of the leading coefficient a n. Example: Find rational zeros of f(x)=x 3 -3x 2 +1
Let f be a polynomial function of degree n≥1 with a positive leading coefficient. Suppose f(x) is divided by x-k using synthetic division. If k≥0 and every number in the last line is nonnegative (positive or zero), then k is an upper bound for the real zeros of f. If k≤0 and the numbers in the last line are alternately nonnegative and nonpositive, then k is a lower bound for the real zeros of f.
Lets establish that all the real zeros of f(x)=2x 4 -7x 3 -8x 2 +14x+8 must lie in the interval [-2,5]
Establish bounds for real zeros Find the real zeros of a polynomial functions by using the rational zeros theorem to find potential rational zeros Use synthetic division to see which potential rational zeros are a real zero Complete the factoring of f(x) by using synthetic division again or factor.
In the 17 th century, mathematicians extended the definition of √(a) to include negative real numbers a. i =√(-1) is defined as a solution of (i ) 2 +1=0 For any negative real number √(a) = √|a|*i
a +bi, where a, b are real numbers ◦ a+bi is in standard form
Z = a+bi = a – bi When do we need to use conjugates?
ax 2 +bx+c=0
Try: x 2 -5x+11=0
Find all zeros: f(x) = x 4 + x 3 + x 2 + 3x - 6
HW: Pg #2-10e, 28-34e
Fundamental Theorem of Algebra – A polynomial function of degree n has n complex zeros (real and nonreal). Linear Factorization Theorem – If f(x) is a polynomial function of degree n>0, then f(x) has n linear factors and F(x) = a(x-z 1 )(x-z 2 )…(x-z n ) Where a is the leading coefficient of f(x) and z 1, z 2, …, z n are the complex zeros of the function.
Suppose that f(x) is a polynomial function with real coefficients. If a+bi is a zero of f(x), then the complex conjugate a-bi is also a zero of f(x)
What can happen if the coefficients are not real? 1. Use substitution to verify that x=2i and x=-i are zeros of f(x)=x 2 -ix+2. Are the conjugates of 2i and –i also zeros of f(x)? 2. Use substitution to verify that x=i and x=1-i are zeros of g(x)=x 2 -x+(1+i). Are the conjugates of i and 1-i also zeros of g(x)? 3. What conclusions can you draw from parts 1 and 2? Do your results contradict the theorem about complex conjugates?
Given that -3, 4, and 2-i are zeros, find the polynomial: Given 1, 1+2i, 1-i, find the polynomial:
The complex number z=1-2i is a zero of f(x)=4x 4 +17x 2 +14x+65, find the remaining zeros, and write it in its linear factorization.
3x 5 -2x 4 +6x 3 -4x 2 -24x+16
HW: Pg. 246 #19-30
F and g are polynomial functions with g(x)≠0. the functions: ◦ R(x)=f(x)/g(x) is a rational function ◦ Find the domain of : f(x)=1/(x+2)
(a) g(x)=2/(x+3) (b) H(x)=(3x-7)/(x-2)
Find horizontal and vertical asymptotes of f(x)=(x 2 +2)/(x 2 +1) Find asymptotes and intercepts of the function f(x)=x 3 /(x 2 -9)