I need help with a calculus problem

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Mathematics

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Instructions: 1) You may use a calculator. 2) You may consult other sources, including other persons, but the work you submit must be substantially your own. 3) You must show your work to receive partial credit. 4) You may lose credit for not showing your work. 5) You must print out this assignment and submit a paper copy by the due date above. 6) Please clearly indicate your answers by circling them. 1. Suppose p(x) = x2 + 8x + 6. Write the expression p(h + 1) βˆ’ p(1) as a sum of terms, each of which is a h constant times a power of h. A) 10 - h B) 10 + h C) –6 + h D) –6 - h 2. Find all real numbers x such that: π‘₯ 4 βˆ’ 53π‘₯ 2 + 196 = 0 3. Find a number b such that 6 is a zero of the polynomial p defined by p(x) = –36 + bx - 6x2 + x3. A) 36 B) –36 C) 6 D) –6 4. Find a polynomial 𝑝 of degree 3 such that βˆ’3, 1 and 4 are zeros of 𝑝 and 𝑝(0) = 4 . 5. If p(x) = a2x2 + 4x + 5 and q(x) = (1 - 2a)x2 - 8x - 5, find a real number a such that (p + q)(x) has degree 1. A) 4 B) -4 C) 0 D) 1 6. Find all choices of (b, c, d) such that 4 and 1 are the only zeros of the polynomial p(x) = x3 + bx2 + cx + d. 7. Factor x24 – y12 completely. 8. Write the domain of the function r(x) as a union of intervals. 5 x 7 βˆ’ 6 x5 βˆ’ 3 r ( x) = x2 βˆ’ 5 A) C) ( βˆ’ο‚₯, βˆ’ 5 ) οƒˆ ( ( βˆ’ 5, 5 ) 5, ο‚₯ ) B) D) 9. Find two distinct numbers x such that t ( x) = ( βˆ’ο‚₯, βˆ’ 5 ) οƒˆ ( βˆ’ ( βˆ’ο‚₯, βˆ’ 5 ) οƒˆ ( βˆ’ ) 5)οƒˆ( 5, 5 5, 5, ο‚₯ ) 1 x + 17 , where t ( x) = 2 . 2 x + 16 x2 R( x) in the form G ( x) + , where q is the denominator of the given x+9 q( x) expression and G and R are polynomials with deg R < deg q. 10. Write the expression 11. Find a constant c such that f (10100 ) ο‚» 4 , where f ( x) = A) 4 B) 16 C) 0 D) -4 cx3 βˆ’ 20 x 2 + 12 x βˆ’ 16 . 4 x3 βˆ’ 8 x 2 + x βˆ’ 3 12. Find the asymptotes of the graph of the function r ( x) = 13. Evaluate the indicated expression. log25 56 A) 6 14. B) 0.33 C) 3 D) 12 Evaluate the indicated expression. log 7 A) 3/2 B) -3/2 C) 2/3 1 343 D) -2/3 15. Find a number t such that log 4 t = βˆ’4 . A) 256 B) 0.0039 C) –256 D) –0.0039 x +8 . x βˆ’ 17 x + 72 2 16. Find a number x such that log5 (8 x + 3) = 3 . Round your answer to two decimal places. 17. Find a number t such that 10t βˆ’ 1 = 0.2 . Round your answer to four decimal places. 10t + 1 18. Find a number x such that 102x + 10x = 30. Round your answer to four decimal places. βˆ’1 19. Find a formula for the inverse function f of the indicated function f. f ( x) = 6 x βˆ’6 βˆ’1 20. Find a formula for the inverse function f of the indicated function f. f ( x ) = 2 οƒ— 9 x βˆ’5 + 3 A) B) 21.  x + 3οƒΆ log  οƒ·  2 οƒΈ + 5 f βˆ’1 ( x) = log 9  x+3οƒΆ log  οƒ·  2 οƒΈ βˆ’5 f βˆ’1 ( x) = log 9 Find the inverse function. A) B) 7x βˆ’ 3 10 x 7 βˆ’ 10 f βˆ’1 ( x) = 3 f βˆ’1 ( x) = C) D)  x βˆ’3οƒΆ f βˆ’1 ( x) = log  οƒ· βˆ’ log 9 βˆ’ 5  2 οƒΈ  x βˆ’3οƒΆ log  οƒ·  2 οƒΈ +5 f βˆ’1 ( x) = log 9 f ( x) = log7 (10 x βˆ’ 3) C) f βˆ’1 ( x) = 10 οƒ— 7 x βˆ’ 3 D) f βˆ’1 ( x) = 7x + 3 10 22. Find a formula for (f  g)(x) assuming that f ( x) = 41+7 x and g ( x) = log 4 x . 23. Find all such numbers x such that log_2(3x + 4) = 4 holds.
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