College Algebra Exercises Section 3.3

As you study these exercises, move your cursor over the light bulbs and the highlighted words.
Study with an active cursor

The definition that pops up when you move the cursor over a highlighted word is what should pop into your mind when you read, hear, or speak that word. The material that pops up when you move the cursor over a light bulb is a suggested strategy for solving the problem. You should always formulate a similar strategy when you attempt to solve a problem. When you move the cursor over an explanation of a step in a solution, the property that pops up is the mathematical justification for taking that action. You must always be able to provide such a justification for every step you take in mathematics.

6) Use long division to find the quotient and remainder when 5x2 -17x -12 is divided by x - 4.
Solution:      
What can we conclude from this?

This process has revealed the quotient 5x + 3 and remainder 0 as assured by the Division Algorithm.
Now we know      5x2 - 17x -12 = (x - 4)(5x + 3)
With this factorization and the Zero Factor Property we can easily determine that the zeros of the function f whose rule is f(x) = 5x2 - 17x -12 are 4 and .
Since both of these zeros are real numbers, they are x-intercepts of the function f.
The graph of the function f is clearly a parabola which opens up and crosses the x-axis at 4 and
All that remains to completely determine the graph of f is to compute the vertex using the formula
Remark: For a quadratic function we do not go to all this effort. It is done here only as an example of what can be interpreted from a division problem.

8) Use long division to find the quotient and remainder when 6x3 - 16x2 + 17x - 6 is divided by 3x - 2.
Solution:      
What can we conclude from this?

This process has revealed the quotient 2x2 - 4x + 3 and remainder 0 as assured by the Division Algorithm.
Now we know   6x3 - 16x2 + 17x - 6 = (3x - 2)(2x2 - 4x + 3)
With this factorization and the Zero Factor Property we can easily determine that one of the zeros of the function f whose rule is f(x) = 6x3 - 16x2 + 17x - 6 is . Since this zero is a real number, it is an x-intercept of the graph of f.
The other two zeros can be found by solving 2x2 - 4x + 3 = 0 with the quadratic formula.

Remark: We now know one x-intercept.
When we use the quadratic formula to solve the quadratic equation 2x2 - 4x + 3 = 0 we find two complex solutions because the expression ( the discriminant) under the radical is b2 - 4ac = 16 - 24 = - 8 < 0.
These complex zeros cannot represent x-intercepts.
Therefore the cubic function whose rule is f(x) = 6x3 - 16x2 + 17x - 6 has only one x-intercept.
Even when we consider the fact that the leading term dominates when far from the origin, we do not have very much information about the graph of f.
Attempts to graph this function even if many points are plotted may be far from correct.
A computer, calculator, or calculus are needed to sketch the graph of f with any degree of assurance that the graph is accurate. A computer generated graph of f is shown at the right.

9) Use long division to find the quotient and remainder when x4 + 5x3 + 6x2 -x - 2 is divided by x+ 2.
Solution:      
What can we conclude from this?

This process has revealed the quotient x3 + 3x2- 1 and remainder 0 as assured by the Division Algorithm.
Now we know   x4 + 5x3 + 6x2 -x - 2 = (x+2)(x3 + 3x2- 1)
With this factorization and the Zero Factor Property we can easily determine that one of the zeros of the function f whose rule is f(x) =  x4 + 5x3 + 6x2 -x - 2 is -2. Since this zero is a real number, it is an x-intercept of the graph of f.
The other three zeros can be found by solving x3 + 3x2- 1 = 0.
Note that now we are faced with finding the zeros of the function g whose rule is g(x) = x3 + 3x2- 1.
Because g is a factor of f, the zeros of g are zeros of f.

Remark: We now know one x-intercept.
When we use the Rational Zeros Rule to solve the quadratic equation x3 + 3x2- 1 = 0 we find there are no rational solutions to the equation x3 + 3x2- 1 = 0.
Consequently the function f has one rational zero. Any other x-intercepts must be irrational.
Note that
there must be at least one more x-intercept because the cubic x3 + 3x2- 1 = 0 must have at least one real solution.
A computer, calculator, or calculus are needed to sketch the graph of f with any degree of assurance that the graph is accurate. A computer generated graph of f is shown at the right.
This graph shows the graph of f as well as the graph of g.
Observe the relationship of the x-intercepts of the two graphs.

42) Use long division to find the quotient and remainder when x3 + 2x2 - 5x - 4 is divided by x+ .
Solution:      
What can we conclude from this?

This process has revealed the quotient and remainder 6 as assured by the Division Algorithm.
Now we know  
This division does not shed any light on the characteristics of the graph of the function f whose rule is f(x) = x3 + 2x2 - 5x - 4
A computer generated graph of f is shown at the right.
Remark: If f has a rational zero it must be an integer (because the leading coefficient is 1).
Looking at the computer generated graph convinces us that none of the three zeros are integers. However, there are three real zeros. They must be irrational numbers. They can be found only by using approximation techniques far beyond the scope of this course. Such techniques are provided on graphing calculators.

  

73) Use long division to find the quotient and remainder when x4 + 6x3 + 11x2 + 6x is divided by x2 +3x+ 2.
Solution:      
What can we conclude from this?

This process has revealed the quotient x2 + 3x and remainder 0 as assured by the Division Algorithm.
Now we know   x4 + 6x3 + 11x2+ 6x = (x2 + 3x)(x2 + 3x + 2)
We can factor each of these quadratic polynomials so that we get
x4 + 6x3 + 11x2+ 6x = (x2 + 3x)(x2 + 3x + 2) = x(x + 3)(x+ 1)(x + 2)
With this factorization and the Zero Factor Property we can easily determine that the zeros of the function f whose rule is
f(x) = x4 + 6x3 + 11x2+ 6x   are 0, -1, -2, and -3
Since each of these zeros are real numbers, they are x-intercepts of the graph of f.

A computer generated graph of f is shown at the right.


EXTRA) Use long division to find the quotient and remainder when is divided by .
Solution:      
What can we conclude from this?

This process has revealed the quotient and remainder as assured by the Division Algorithm.
Now we know  
This division does not shed any light on the characteristics of the graph of the function f whose rule is
A computer generated graph of f is shown at the right. Note the numbers on the range (the f(x) axis).
The graph below is a computer generated graph of f which shows more detail in the vicinity of the origin.