Polynomial Division Ex..

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Polynomial Division Ex.

Ex. Divide 6x3 – 19x2 + 16x – 4 by x – 2, and use the results to factor the polynomial completely.

Ex. Divide x2 + 3x + 5 by x + 1.

Ex. Divide x3 – 1 by x – 1.

Ex. Divide 2x4 + 4x3 – 5x2 + 3x – 2 by x2 + 2x – 3.

Ex. Divide x4 – 10x2 – 2x + 4 by x + 3.

Ex. Show that (x – 2) and (x + 3) are factors of f (x) = 2x4 + 7x3 – 4x2 – 27x – 18, then find the remaining factors.

Practice Problems Section 3.3 Problems 5, 13, 15, 19, 57

Zeros of a Polynomial Fundamental Theorem of Algebra If a polynomial has degree n, then it has exactly n zeros. - This includes repeating zeros - Some of the zeros may be complex numbers - If k is a zero, then the polynomial can be divided by (x – k) - We know that the zeros exist, though we may not be able to find them

a) f (x) = x – 2 has one zero, x = 2 b) f (x) = x2 – 6x + 9 has two zeros, x = 3 and x = 3 c) f (x) = x3 + 4x has three zeros, x = 0, x = 2i, and x = -2i d) f (x) = x4 – 1 has four zeros, x = 1, x = -1, x = i, and x = -i

To help us find zeros, we can use: Rational Zero Test If a polynomial has integer coefficients, every rational zero has the form where p = factor of constant coefficient q = factor of lead coefficient

Ex. Factor f (x) = x4 – x3 + x2 – 3x – 6.

Ex. Find all real solutions of 2x3 + 3x2 – 8x + 3 = 0

If a complex number is a zero of a polynomial, then so is the conjugate. Ex. Find a fourth-degree polynomial that has 1, -1, and 3i as zeros.

Ex. Find all zeros of f (x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero.

Ex. Express f (x) = x4 – 3x3 + 6x2 + 2x – 60 as the product of linear factors.

Practice Problems Section 3.4 Problems 1, 7, 11, 17, 23, 37, 47

Mathematical Modeling In the real world, you will have data points and will want to find a function to describe the situation.

Ex. Use your calculator to find a linear regression for the data below.

y is directly proportional to x if y = kx for some constant k k is called the constant of proportionality We also say varies directly

Ex. The state income tax is directly proportional to gross income Ex. The state income tax is directly proportional to gross income. If the tax is $46.05 for an income of $1500, write a mathematical model for income tax.

y is directly proportional to the nth power to x if y = kxn for some constant k Ex. The distance a ball rolls down a hill is directly proportional to the square of the time it rolls. During the first second, the ball rolls 8 ft. Write an equation for this model. How long does it take for the ball to roll 72 ft?

y is inversely proportional to x if for some constant k We also say varies inversely z is jointly proportional to x and y if for some constant k We also say varies jointly

Ex. The simple interest for a savings account is jointly proportional to the time and the principal. After one quarter, the interest on a principal of $5000 is $43.75. Write an equation for this model.

Practice Problems Section 3.5 Problems 7, 33, 63, 65, 67