College Algebra Unit 3

(h,k) means

(x,y) of vertex

Multiplicities of zeros examples

f(x) = (x-3)²*²(x+2)³(x+7) Set each factor equal to 0 and solve ²*²√(x-3)²*² = ²*²√0 = x-3 = 0 = x = 3 Multiplicity 4 is even, so the graph touches at x = 3

quadratic function

f(x) = a(x-h)²+k

Quadratic function

f(x) = ax²+bx+c, where a ≠ 0, is a polynomial function of degree 2

Constant function

f(x) = c, where c ≠ 0, is a polynomial function of degree 0

Linear function

f(x) = mx+b, where m ≠ 0, is a polynomial function of degree 1

i² is the same as

-1

How to use synthetic division

1. Arrange the polynomial in descending powers, with a 0 coefficient for any missing term 2. Write c for the divisor, x-c. To the right, write the coefficients of the dividend 3. Write the leading coefficient of the dividend on the bottom row 4. Multiply c times the value just written on the bottom row. Write the product in the next column in the second row 5. Add the values in this new column, writing the sum in the bottom row 6. Repeat this series of multiplications and additions until all columns are filled in 7. When writing all the values out as a new expression, remember the degree of the first term is one less than the original. The remainder is the final value on the far right bottom row

How to use long division with polynomials

1. Arrange the terms of the dividend and the divisor in descending powers of any variable 2. Divide the first term in the dividend by the first term in the divisor. The result is the first term in the quotient 3. Multiply every term in the divisor by the first term of the quotient. Write the resulting product beneath the dividend with like terms lined up 4. Subtract the product from the dividend 5. Bring down the next term in the original dividend and write it next to the remainder to form a new dividend 6. Use this new expression as the new dividend as needed

How to tell if any of the possible rational zeros are rational zeros of the polynomial function

1. Create a p/q list of possible rational zeros 2. Graph the function in your calculator to see the x-intercepts 4. If any of the x-intercepts appear in the list, use synthetic division to check it out. If the remainder is 0, then you have found a solution. Repeat this step as needed to make the equation smaller 5. Remember that is a graph touches the x-axis at a x-intercept and turns around, that multiplicity is even. If the graph crosses the x-axis at a x-intercept, the multiplicity is odd

How to graph f(x) = a(x-h)²+k

1. Figure out if the graph opens up or down. If a>0, it opens upward. If a<0, it opens downward 2. Determine the vertex, (h,k), of the parabola 3. Find any x-intercepts by solving f(x)=0. The function´s real zeros are the x-intercepts 4. Find the y-intercept by computing f(0) 5. Plot the intercepts, the vertex, and additional points as necessary. Connect the dots with a parabolic curve 6. A dashed line can used to show the axis of symmetry

Descartes' Rule of Signs

1. The number of positive real roots is either a. the same as the number of sign changes of f(x) b. or less than the number of sign changes of f(x) by a positive even integer. If f(x) has only one variation in sign, then f has one positive real zero 2. The number of negative real roots is either a. the same as the number of sign changes of f(-x) b. or less than the number of sign changes of f(-x) by a positive even integer. If f(-x) has only one variation in sign, then f has exactly one negative real zero

How to graph polynomial functions

1. Use the Leading Coefficient Test to figure out the graph´s end behavior 2. Find x-intercepts by setting f(x) = 0 and solving the resulting polynomial equation. If there is an x-intercept at r as a result of (x-r)k in the total factorization of f(x), then a. if k is even, the graph touches the x-axis and turns around b. if k is odd, the graph crosses the x-axis at r c. if k>1, the graph flattens out near (r,0) 3. Find the y-intercept by computing f(0) 4. Use symmetry, if applicable, to help draw the graph: a. y-axis symmetry: f(-x) = f(x) b. origin symmetry: f(-x) = -f(x) 5. Use the fact the maximum number of turning points on the graph is n-1, where n, the degree of the polynomial function, to check whether it´s drawn correctly

Axis of symmetry

A line that divides a graph into two congruent reflected halves

Linear Factorization Theorem

An nth degree polynomial can be expressed as the product of a nonzero constant and n linear factors, where each linear factor has a leading coefficient of 1. Just like a regular number can be broken down into its prime factorization, polynomials can also be factored down to its prime factorization to where the degree of each factor is 1. This is called linear factorization

When the divisor is a monomial...

Divide each term of the dividend by the monomial and simplify the resulting expressions

Rational Zero Theorem

If P(x) is a polynomial with integer coefficients and if is a zero of P(x) (P( ) = 0), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x) To use the theorem, list all the factors of the constant (the final value of the equation) and place them in the numerator. List all the factors of the leading coefficient in the denominator. Divide all each numerator value by each denominator value separately. Remember to write ± in front of the list of resulting answers

Properties of Roots of Polynomial Equations

If a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots If a+bi is a root of a polynomial equation with real coefficients (b≠0), then the imaginary a+bi is also a root. Imaginary roots, if they exist, occur in conjugate pairs

Remainder Theorem

If a polynomial f(x) is divided by x-c, then the remainder is f(c)

Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n where n≥1, then the equation f(x)=0 has at least one complex root. Basically, the degree of the polynomial equation is the same as the number of roots to the polynomial equation

Multiplicities and x-intercepts

If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether the multiplicity is even or odd, graphs tend to flatten out near zeros with multiplicities greater than one

Leading Coefficient Test

If the leading coefficient is positive, the right end of the line will rise If the leading coefficient is negative, the right end of the line will fall If the leading coefficient is odd, the ends will go in opposite directions If the leading coefficient is even, the ends will go in the same direction

Intermediate Value Theorem

If two points on a graph have y-values with opposite signs, there must be an x-intercept between two points

Factor Theorem

Let f(x) be a polynomial. If f(c)=0, then (x - c) is a factor of f(x) If (x - c) is a factor of f(x), then f(c)=0

Multiplicities

Multiplicities are the same things as roots or solutions. They are x-intercepts on graphs

Multiplicities of zeros

Once a polynomial function has been factored, we can use the number of times a factor has been used to determine if the graph will cross or touch the x-axis at a given x-intercept. In factoring the equation, if the same factor x-r, where r is an x-intercept, occurs k times (and no more), we call r a zero with multiplicity k

How to tell if equations are polynomial functions

Polynomial functions do not have variable exponents that are negative or fractional (like square roots). All exponents must be non-negative integers Example of non-polynomial function: h(x) = 8√x+4x²-9

Smooth, continuous graphs

Polynomial functions of degree 2 or higher have smooth graphs (only rounded corners with no sharp corners) and continuous (no breaks or gaps)

Leading coefficient

The coefficient of the term with the highest power

Turning points of polynomial functions

Turning points are distinct points where a graph changed direction. If a polynomial function has degree n, then the graph has at most n-1 turning points. The y-coordinate of each turning point will represent a relative minimum or relative maximum Example: f(x) = x³-6x²+8x+1. Degree n is 5, so there is at most 5-1 or 4 turning points in the graph

Leading Coefficient Test examples

f(x) = x²-4x² Leading coefficient 1 is positive, so the right end rises. Degree 2 is even, so the left end does the same and rises on the left g(x) = -x³+2x³ Leading coefficient -1 is negative, so the right end falls. Degree 3 is odd, so the left end does the opposite and right to the left

√-1 is the same as (and is always rewritten as)

i

Division Algorithm

if f(x) and d(x) are polynomials such that d(x) is not equal to 0, and the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that f(x) = d(x)q(x) + r(x). The remainder, r(x), equals 0 or it is of degree less than the degree of d(x). If r(x)=0, we say d(x) divides evenly into f(x) and that d(x) and q(x) are factors of f(x)