15. Problems: Algebra Test

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15. Problems: Algebra Test [ Back ] [ Area Intro ] [ Next ]

Problems on Quadratics, Straight Lines, Polynomials, Logs and Exponentials

1. (10 points) Express A and B as simplified, improper fraction: For the calculation of A, cancel common factors in numerators and denominators before multiplication. For the calculation of B, use the least common denominator.

A =

25

27

.

81

100

B =

3

26

+

7

10

For A and B, there is 1 point for right answer and 4 points for following instruction on how to arrive at the answer. Show work! If you use a calculator (not recommended), show the reasoning needed to arrive at your solution without its use.

2. (10 points) Use algebra to find the intersection point of the two lines

y = ¾x - ¼ y = ¼x + 2

3. (4 points) (a) Find the slope of the straight line K if K is perpendicular to a line L with slope 2.
(6 points) (b) Find the slope and an equation for a straight line T which passes through [3,5] and [8, 20].
Express the equation in slope intercept form y= m x + b

4. (10 points) Compute product (x⁴+3x³+6x² + 3x + 1)(x² + 2x + 1) using a column method.

5. (10 points) Use the polynomial long division to find polynomials a(x) (the quotient) and r(x) (the remainder) such that

x⁴+3x³+6x² + 4x + 2= a(x)(x² + 2x + 1) + r(x)

with degree (r(x)) < 2.

6. (2 points) (i) Use the ln(x) button on your calculator to fill in the table (or a copy of it)

x	1/6	0.2	0.25	1/3	0.5	1	2	3	4	5	6
ln(x)						0

(3 points) (i) Use the e^x button on your calculator to complete in the table (or a copy of it).

x	-1.792	-1.61	-1.39	-1.099	-0.693	0	0.693	1.099	1.39	1.61	1.79
exp(x)						1

(5 points) (iii) Draw the straight line y = x from [-2,-2] to [6,6] on graph paper. Then on the same graph, use the above points to sketch graph of y = ln(x) for x in the interval [1/6, 6] and y = exp(x) = e^x for x in the interval [-1.79, 1.79]. Label all curves.

7. (3 points) (i) Solve x²-5x+ 4 =0 using factorization by inspection. Show all ways for 4 to equal the product AB of two integers A and B.
    (3 points) (ii) Solve x²-5x+4 = 0 with the quadratic formula. Show work.
    (4 points) (iii) Solve x²- 5x + 4 =0 starting with the method of completing the square. Show work.

8.  (4 points) (i) Find the x- and y-intercepts of the quadratic y = x²-5x+4 and the straight line y = -2x + 8 with the x- and y-axes. Hint: See 7(ii) or (iii).
     (3 points) (ii) Sketch the curves y = x²-5x+4 and y = -2x + 8. Identify the axis of symmetry for the quadratic. Label axes and all intercepts.
     (3 points) (iii) Find the coordinates of the intersection points for the quadratic y = x²-5x+4 and line y = -2x + 8. Show reasoning. Hint: x²-3x-4 = (x-4)(x+1).