College Algebra Final Exam

If polynomial p(x) of degree n>1 is divided by (x-a) where a is a constant, then the remainder is p(a)

find -b/2a and than plug back in to equation

synthetic division, graphing, factoring

Percentage Profit = ((selling price-cost price)/(cost price))*100

Monomial- 5x2 Binomial- 2x2+5x Trinomial- 5x2+3x+6

This relationship is always true: If a polynomial has rational roots, then those roots will be fractions of the form (plus-or-minus) (factor of the constant term) / (factor of the leading coefficient). However, not all fractions of this form are necessarily zeroes of the polynomial. Indeed, it may happen that none of the fractions so formed is actually a zero of the polynomial. Find all possible rational x-intercepts of x4 + 2x3 - 7x2 - 8x + 12. The constant term is 12, with factors of 1, 2, 3, 4, 6, and 12. The leading coefficient in this case is just 1, which makes my work a lot simpler. The Rational Roots Test says that the possible zeroes are at: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved ± 1, 2, 3, 4, 6, 12 = -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12

The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph. The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. So, the sign of the leading coefficient is sufficient to predict the end behavior of the function.

Polynomial End Behavior: 1. If the degree n of a polynomial is even, then the arms of the graph are either both up or both down. 2. If the degree n is odd, then one arm of the graph is up and one is down. 3. If the leading coefficient an is positive, the right arm of the graph is up. 4. If the leading coefficient an is negative, the right arm of the graph is down.

To predict the end-behavior of a polynomial function, first check whether the function is odd-degree or even-degree function and whether the leading coefficient is positive or negative.

Turning Points A Turning Point is an x-value where a local maximum or local minimum happens:

Never more than the Degree minus 1

The Degree of a Polynomial with one variable is the largest exponent of that variable.

Graphs will be continuous and smooth Even exponents behave the same: above (or equal to) 0; go through (0,0), (1,1) and (-1,1); larger values of n flatten out near 0, and rise more sharply. Odd exponents behave the same: go from negative x and y to positive x and y; go through (0,0), (1,1) and (-1,-1); larger values of n flatten out near 0, and fall/rise more sharply Factors: Larger values squash the curve (inwards to y-axis) Smaller values expand it (away from y-axis) And negative values flip it upside down Turning points: there will be \"Degree-1\" or less. End Behavior: use the term with the largest exponent

The formula for Hooke's Law is \"F = kd\", where \"F\" is the force and \"d\" is the distance. (Note that, in physics, \"weight\" is a force. These Hooke's Law problems are often stated in terms of weight, and the weight is the force.)

\"F varies as x\" means F = kx \"F varies jointly as x and y\" means F = kxy \"F varies as x + y\" means F = k(x + y) \"F varies inversely as x\" means F = k/x

To graph a rational function, you find the asymptotes and the intercepts, plot a few points, and then sketch in the graph.

set the denominator equal to zero to find any forbidden points

If the numerator and denominator have the same degree (they're both linear), the asymptote will be horizontal, not slant, and the horizontal asymptote will be the result of dividing the leading coefficients

Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. (They can also arise in other contexts, such as logarithms, but you'll almost certainly first encounter asymptotes in the context of rationals.)

1. Identify the total number of real or complex zeros (corollary to Fundamental Theorem of Algebra). 2. Identify the possible number of positive, negative, and complex zeros (Descartes' Rule of Signs). 3. List the possible rational zeros (Rational Root Theorem) 4. Try possible rational zeros until you find one that works. 5. After each division by a positive value, check for possible upper bounds. 6. After each division by a negative value, check for possible lower bounds (Upper and Lower Bound Theorems) 7. After you find a possible rational root that actually works, take the quotient and continue to try to factor it until it is down to a quadratic or less. Once it is a quadratic or less, there are other ways to solve it. 8. Write the linear and or linear / irreducible quadratic factorization (next section)

If a polynomial function has integer coefficients, then every rational zero will have the form p/q where p is a factor of the constant and q is a factor of the leading coefficient. Make sure the polynomial has integer coefficients. Multiply to get rid of fractions or decimals if need be (be sure to later divide). This only addresses the rational zeros. There may be real, but non-rational roots. There may be complex roots involving i. This says that rational zeros will have this form. It does not say that everything that has this form is a rational zero. What it does give you is a list of possible rational zeros Example: f(x) = 4x^5 - x^2 + 12 Possible rational zeros will be of the form (factor of 12) over (factor of 4). A division table can help you find all these values

The maximum number of positive real roots can be found by counting the number of sign changes in f(x). The actual number of positive real roots may be the maximum, or the maximum decreased by a multiple of two. The maximum number of negative real roots can be found by counting the number of sign changes in f(-x). The actual number of negative real roots may be the maximum, or the maximum decreased by a multiple of two. Complex roots always come in pairs. That's why the number of positive or number of negative roots must decrease by two. Consider: f(x) = 3x^6 + x^5 - x^4 + 3x^3 + 2x^2 - x + 1. The signs in f(x) are + + - + + - +. There are 4 sign changes (+ to -) or (- to +). Now, f(-x) = 3x^6 - x^5 - x^4 - 3x^3 + 2x^2 + x + 1. The signs in f(-x) are + - - - + + +. There are 2 sign changes (+ to -) or (- to +).

Complex solutions come in pairs. If (a+bi) is a solution, then its complex conjugate (a-bi) is also a solution.

Every polynomial in one variable of degree n, n > 0, has exactly n real or complex zeros.

Every polynomial in one variable of degree n, n > 0, has at least one real or complex zero.

Synthetic Division To divide a polynomial synthetically by x-k, perform the following steps. Setup Write k down, leave some space after it. On the same line, right the coefficients of the polynomial function. Make sure you write the coefficients in order of decreasing power. Be sure to put a zero down if a power is missing. Place holders are very important For now, leave a blank line. Draw the left and bottom portions of a box. The left portion goes between the k and the coefficients. The bottom portion goes under the blank line you left. Synthetic Division Once you have things set up, you can actually start to perform the synthetic division. Bring the first coefficient down to the bottom row (below the line) Multiply the number in the bottom row by the constant k, and write the product in the next column of the second row (above the line). Add the numbers in the next column and write the total below the line. Repeat steps 2 and 3 until all the columns are filled. Interpreting the Results The very last value is the remainder. If the remainder is zero, you have found a zero of the function. The rest of the values are the coefficients of the quotient. Each term will be raised to the one less power than the original dividend. (If it was a fourth degree polynomial to start with, the quotient will be a third degree polynomial).

When a polynomial function f is divided by x-k, the remainder r is f(k). Okay, now in English. If you divide a polynomial by a linear factor, x-k, the remainder is the value you would get if you plugged x=k into the function and evaluated. Now, tie that into what we just said above. If the remainder is zero, then you have successfully factored the polynomial. If the remainder when dividing by (x-k) is zero, then the function evaluated at x=k is zero and you have found a zero or root of the polynomial. Plus, you now have a factored polynomial (the quotient) which is one less degree than the original polynomial. If the quotient is down to a quadratic or linear factor, then you can solve and find the other solutions.

The real (that is, the non-complex) zeroes of a polynomial correspond to the x-intercepts of the graph of that polynomial. So you can find information about the number of real zeroes of a polynomial by looking at the graph, and conversely you can tell how many times the graph is going to touch or cross the x-axis by looking at the zeroes of the polynomial (or at the factored form of the polynomial).

1. Zeros 2. Test Points 3. End Behavior 4. Graph

The amount of local extrema is determined by n - 1, where as N is the highest exponent in the graph. A graph of x4 will have 3 local extrema.