 # Mathematics Questions

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(1 point) Suppose f ( x) = x5 ( 5 - 2x3 ). (a) The roots of f ( x) are x = O, 1.35720880 Is repeated , enter lt only once .) (b) As X -+ 00 , f (x) -+ (b) As X -+ - 00, J (x) -+ help (numbers) (lf a root help (numbers) help (numbers) (1 point) For the tunction y = ( x - 3) (x + 2) ( 4x - 2), its y-i ntercept is y = its X-intercepts are X = Note: lf there is more than one answer enter them separated by commas . lf there are none , enter none . When X -+ 00, y -+ oo ( Input + or - for the answer ) When X -+ -00, y -+ oo ( Input + or - for the answer ) (1 point) Use synthetic division and the Remainder Theorem to evaluate P( e), where P (x) = x3 -3 x2 + 6x -14 , c = 2 The quot ient is [ l (1 point) Use synthetic division and the Remainder Theorem to evaluate P( e), where P( x) = x2 + 8x + 9, e = -1. °The quot ient is (1 point) Use synthetic division and the Remainder Theorem to evaluate P( e), where P(x) = x 4 + 7x3 + 3x 2 + 24x + 22, e= -7 th e quotient is (1 polnt) For the functlon y = l X - ti) l x + O), its y-i ntercept is -48 its X-intercepts are X = 8, -6 Note: lf there Is more than one X- lntercept wrlte the x-values separated by commmas. When X -+ 00, y -+ When X -+ -00, y -+ 00 (Input + or - for the answer) 00 (Input + or - for the answer) (1 polnt) What Is the degree of the followlng polynomlal? (x3a - l)( x3ª + 3) Degree: (1 point) The potynomiat f ( x) = ( 4 - 2x )(3x - 3)2 has degree= -3 ----- teadlng coeff lclent= --18- ---- constant coeff lclent= 1 (1 point) The potynomiat f ( x) = x5 ( 5 - 2x3 ) has degree= 8 teadlng coeff lclent= -2 constant coeff lclent= (1 point) Suppose f(x) = ( 4 - 2x)(3x - 3)2 . (a) The roots of f (X) are X = ¡ 2, 1 J help (numbers) (lf a root ~------~ is repeated , enter it only once .) (b) As X --+ 00, f(x) --+ help (numbers) (b) As X --+ -oo, f (X) --+ ~------~ help (numbers) (1 point) Suppose f(x) = - (3 - x)(x + 2)(5x + 2)2. (a)Therootso f f(x) are x = 3,-2,-215 is repeated , enter it only once.) (b) As X --+ 00, f (x) --+ (b) As X --+ -00, f (x) --+ help (numbers) (lf a root help (numbers) help (numbers) (1 point) Find the quotient and remainder using synthetic division for x5 - x4 + 8x3 - 8x2 + 2x - 10 x-1 The quotien t is (1 polnt) Flnd a degree 3 polynomla l having zeros -3, 4 and 8 and the coefflcle nt of x 3 equal 1. The polynomlal Is (1 point) Ust ali possible rational roots for the function f(x) = 7x4 + 5x3 + Ox2 - 9x + 91. Give your list in increasing order. Beside each poss ible rational root, type "yes· if it is a root and · no· if it is nota root. Leave any unnecessary answer blanks empty. Possible rational root: - Is it a root? 11 1 . Possible rational root: Is ita root? - Possible rational root: Is ita root? - Possible rational root: Is ita root? - Possible rational root: Is ita root? - Possible rational root: Is ita root? - Possible rational root: Is ita root? - Possible rational root: Is ita root? - Possible rational root: Is ita root? - Possible rational root: Is ita root? - Possible rational root: Is ita root? - Possible rational root: Is ita root? (1 point) Find all the real zeros of the polynomial P(x) = x3 - 2x2 - 9x + 4. Enter the real zeros as a comma -separated list: 14,-1 + sqrt(2), -1- sqrt(2) Note: Give exact answers; no decimals. lf needed, enter square root as sqrt, e.g. the square root of 2 should be enterad as sqrt(2). When x --> oo, P( x) --> 1 ---.------""--, When x --> -oo, P( x) --> Note: lf your answer is oo, enter infinity; if your answer is -oo, enter -infinity. (1 point) Find the quotient and remainder using synthetic division for x5 - x4 + 3x3 - 3x2 + 9x - 16 x - 1 The quotient is "'---------- ( 1 polnt) Glven that f ( x) Is a cublc functlon wlth zeros at - 6, 2, and 9, flnd an equatlon for f (x ) given that / (- 5) = 9. f( x) = (1 polnt) Flnd a degree 3 polynomlal whose coefflclent of x3 equal to 1. The zeros of thls polynomlal are 1, - 2i, and 2i. Slmplify your answer so that it has only real numbers as coeffic ients. Your answerIs (1 point) Find a degree 3 poly nomial that has zeros - 2, 3 and 8 and in which the coeffic ient of x 2 is -18. The polynomial is (1 point) For p(x) = 1.5 + x1 + 3x2 + 7x3 + 2x4 + 8x5 + 8x6 Degree of p = Leading term = Leading coefficien 6 8X"6 t = 8 Determine the coefficient and the degree of each term. Term Coefficient Degree 1.5 X 3x2 7x3 2x4 8x 5 8x6 l O A B l O ' ' e o 1. f(x) = (x - l)(x - 3)(x - 5) 2. f(x) = (x - l)(x - 3)(x + 5) 3. f(x) = (x - l)(x + 3)(x + 5) 4. f(x) = (x + l )(x + 3)(x + 5) . . 1 /1 (\ • . • • .. • ... J . . . . \ A B I , . • ' ' J• V . e D e 1.f( x) = - x3 + x2 + 6x a 2. f( x) = - x3 - x2 + 6x d 3 . f ( x) = x3 - x2 - 6x b 4 . f ( x) = x 3 + x 2 - Gx [ ••• A B •• • •• e D d 1. J(x) = x4 + 5x2 + 4 a 2.f(x) = x4 - 5x2 +4 b 3. J(x) = -x 4 + 5x2 - 4 e 4.f(x) = -x 4 -5x 2 - 4 (1 point) For the tunction y = ( x - 2) (x + 2) (6x - 2), its y-i ntercept is y = its X-intercepts are X = Note: lf there is more than one answer enter them separated by commas. lf there are none, enter nona . When X -+ 00, y -+ oo ( Input + or - for the answer ) When X -+ -00, y -+ oo ( Input + or - for the answer ) (1 point) For the tunction y = x4 - 5x 3 - 14x2, its y-i ntercept is y = .._ _______ ___, its X-intercepts are X = Note: lf there is more than one answer enter them separated by commas. lf there are none, enter nona . When :t -+ oo, y -+ oo (Input + or - tor the answer) When :t -+ -oo, y -+ oo (Input + or - tor the answer) (1 point) Let f(x) = x2 + 2x + 3 and g(x) = 4x3 - 5x2 + 4x + 1 For brevity let's write deg(p) for the degree of a polynomial p. So in the above example, deg(f) = 2 and deg(g) = 3. You can answer the follow ing questions without actually computing the indicated functions . deg(f + g) deg(f - g) deg(f g) deg(f o g) deg(g o f) (1 point) For the function y = ( x - 1 )2( x - 6), its y-intercept is y = its X-i ntercepts are X = Note: lf there is more than one answer enter them separated by commas. lf there are nona, enter nona . When X --> 00, y --> oo ( Input + or - for the answer ) When X --> -00, y --> oo ( Input + or - for the answer ) (1 point) For the tunction y = x 3 - x 2 - 12x, its y-i ntercept is y = its X-i ntercepts are X = Note: lf there is more than one answer enter them separated by commas. lf there are none, enter nona . When X -+ 00, y -+ oo ( Input + or - for the answer ) When X -+ -00, y -+ oo ( Input + or - for the answer ) (1 point) For the tunction y = x 4 - 5x3 - 24x2, its y-i ntercept is y = its X-i ntercepts are X = Note: lf there is more than one answer enter them separated by commas. lf there are none, enter nona . When X -+ 00, y -+ oo (Input + or - tor the answer) When X -+ -oo, y -+ oo (Input+ or - tor the answer) (1 point) For the tunction y = x3 - x 2 - 12x, its y-i ntercept is y = its X-i ntercepts are X = Note: lf there is more than one answer enter them separated by commas. lf there are none, enter nona . When X -+ 00, y -+ oo ( Input + or - for the answer ) Whenx -+ -oo, y-+ oo ( Input + or - for the answer ) (1 point) For the function y = x( X - 7) (X+ 6), its y-intercept is y = its X-i ntercepts are X = Note: lf there is more than one answer enter them separated by commas. lf there are none, enter none . When X -+ 00, y -+ oo ( Input + or - for the answer ) When X -+ -00, y -+ oo ( Input + or - for the answer ) (1 polnt) The polynom lal of degree 4, P( x) has a root of multlpllcity 2 at :i; = 4 and roots of multip licity 1 at X = O and X = -4. lt goes through the point ( 5, 4.5} Find a formula for P( X). P(x) = (1 point) Find the quotient and remainder using long dlvision for 2x3 - 8x2 + 7 x - 13 2x2 + 5 The quotlent Is

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