Lycée des arts Math Grade Algebraic expressions S.S.4

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1 Lycée des arts Math Grade lgebraic expressions S.S.4 Exercise I: Factorize each of the following algebraic expressions, and then determine its real roots. (x) = 4x (3x 1) (x + ) (3x 1) + 3x 1 B (x) = (x + 1) (4x + 3) 5x (4x + 3) + (x 1) (4x + 3) (x) = (x ) (5x 1) + (x 3) (5x 1) (3x + ) (5x 1) (a) = (3a + 1) (a + 1) + a 1 E (y) = 4y 9 + (y + 3) (y 5) F (x) = (x 3) (x + 7) + (x 6) (3x 1) (9 3x) (x + 1) G (x) = (4x 3) ( x + 5) + (x 1) (x 5) + (x 5) ( x + 5) H (x) = 6(x 16) (3x + 1) (x 4) + (8 x) (x + ) I (x) = (x + 7) (3x + 4) + (9x + 4x + 16) J (x) = 3x 1 + (x 4) ( x) (x 4x + 4) K (x) = (6x 1x + 6) + (3x 3) (x 1) (x + 1) L (x) = 4x 4x + 1 (1 x) (3x + 5) 1x + 3 M (a) = (3a ) (a + 1) (3a ) N (x) = x (x 1) x (x + 1) Exercise II: Factorize each of the following expressions: = a 3 + a + a + 1 ; B = xy 3x y + 6 = b a + ab ; = 10xy + 4x 5y E = x y + a + b + xy ab ; F = 5(3x y) 16(5x + 3y) G = 3(5x 1) 3(x + ) ; H = (a 3) (a 3) + 1 I = (3x + ) + (3x + ) (x 1) + (x 1) ; J = (a + 4) (a + 4) (a + 1) + (a + 1) K = 4x 4xy + y 9x y ; L = 5(x 4) x + 4x 4 + (6 3x) (x + 3) M = 3(x 1) + x (x 1) (x + ) ; N = 5x + (5x 3) (x + 7) 9 O = (3x 5) 4x 4 ; P = (4a + 1) (5a ) Q = a ab b 1 ; R = x 4x + (x 4) x 1 S = 4 3 5x 4 3 x ; T = October - Math S.S-4 lgebraic expressions Grade 9 1/5

Exercise III: 1 3 1. 1 3 3 1 1 x x x x B x x x x x. 1º) a) Expand and reduce x and b) Factorize x and B x. B x. º) In the adjacent figure we have: * B is a rectangle such that = 3x cm.

2 Exercise III: x x x x B x x x x x. 1º) a) Expand and reduce x and b) Factorize x and B x. B x. º) In the adjacent figure we have: * B is a rectangle such that = 3x cm. * EFG and PMNB are two squares such that: E = x and PM = 1 cm. (x is a real number expressed in cm) a) Show that x 3x. Frame the value of x. b) Express, in terms of x, the area of the rectangle B. x represents the area of the shaded region. c) educe that d) Is there any value of x, for which the area of B is the triple of that of the shaded region? Justify. Exercise IV: (x) = (x 3) (x 1) + 4(3 x) and B (x) = x 3 3x x º) a) Expand and reduce (x). b) alculate ( 3). What can you conclude? c) Solve the equation: B (x) = 3 3x. º) a) Factorize (x), and then show that B (x) = (x 3) (x 1) (x + 1). b) Solve: (x) = B (x). () x 3º) onsider the fractional expression: F () x B () x a) etermine the domain of definition of F (x). b) Simplify F (x), and then calculate F ( ). c) Solve F (x) = 0 and F (x) = 1. Exercise V: Given: (x) = 4x x 4x and B (x) = (5 x) (4x 1). 1º) a) Write (x) and B (x) as products of factors of the first degree. b) For what value of x, the double of (x) is the triple of B (x)? Justify. º) a) etermine the prohibited of values of b) Simplify f (x), and then reduce f. c) Solve : f () x and f () x 0. 3 f () x () x. B () x October - Math S.S-4 lgebraic expressions Grade 9 /5

3 Exercise VI: and P x x a b x a x a x b bx c a Q x x x R x 3m 1 x 5m x m. a, b, c and m are four real numbers m is a parameter. 1º) alculate m knowing that «3» is a root of R x. º) a) Show that: P x 3 b x 3a b x c 3. b) alculate a, b and c so that the polynomials P x and c) alculate a, b and c so that the polynomial P x Exercise VII: Q x are identical. is identically null. Given : P (x) = (6x 3) (x 1) 3(4x 1) + 3(x 1) and Q (x) = P (x) 5x (x -3) 1º) a) Expand and reduce P (x). b) Solve the equation P (x) = 9. º) a) Show that: P (x) = 3(x 3) (x 1), and then solve P (x) = 0. b) Verify that Q (x) is a perfect square. c) educe the solutions of the equation Q (x) = 4. 3º) In this part x represents a real number greater than 1 (x > 1). We designate by S the area of the rectangle B where B = x 1 and B = x 1, and S the area of the triangle IJK right at I where IJ = x 1 and IK = x + 1, and S is that of the square MNPQ of side MN = x 1. a) Show that: 3S 6S + 3S = P (x). b) Find the value of x that verifies the equation: S + S = S. Exercise VIII: Given the following algebraic expressions. f (x) = (x ) + 4(x 5) and g (x) = (x ) 1(x 5). 1º) Expand and reduce f (x) and g(x). º) Verify that: f (x) = x 4. x 4 and g (x) = x 8 3º) Solve: f (x) = 0; g (x) = 0 and g (x) = 5. x - 4º) In what follows, x is expressed in cm such that x > 5. E F The adjacent figure shows a square B of side x - 5 B = x and a rectangle EFG of dimensions of B and EFG a) Express S (x) and S (x) in terms of x. G = 4 and FG = x 5. x - We designate by S (x) and S (x) the respective areas G b) Solve each of the following equations: S (x) = 3S (x) et S (x) + S (x) = 0. x - B October - Math S.S-4 lgebraic expressions Grade 9 3/5

Exercise IX: Given: E(x) = (x + 3) (x + 1) (x + ) and P(x) = (x + 1) (x + 3) (x 3). 1º) Expand and reduce E(x) and P(x). º) a) Use the reduced of E(x) to calculate S = 88891 (88889) (88890).

4 Exercise IX: Given: E(x) = (x + 3) (x + 1) (x + ) and P(x) = (x + 1) (x + 3) (x 3). 1º) Expand and reduce E(x) and P(x). º) a) Use the reduced of E(x) to calculate S = (88889) (88890). b) Use the reduced of P(x) to calculate T = Exercise X: The unit of length is the cm, x is a real number such that x > º) Express, in terms of x, the area of the rectangle B that we denote P(x). º) Express, in terms of x, the area of the rectangle IBJ that we denote Q(x). 3º) Find the value of x for which we have P(x) = x. 4º) etermine the value of x for which the rectangles B and IBJ have two equal areas. x + 3 I B 5º) onsider the expression: F () x P () x x - 1. Q () x a) For what values of x, is F (x) defined? b) Simplify F (x), and then calculate F ( ). c) Solve the equations: F (x) = 0 and F (x) = 3. Exercise XI: In this exercise the unit of length is the cm. Let x be a real number such that x > 6. In the attached figure we have: x - B 1 E 1 * B is a rectangle such that B = x - * BEFG is a square of side 1. G 1 F * G is a point on [B] such that G = x º) Express the area of B in terms of x. º) Show that the area of the polygon EFG is expressed by (x) = (x )(x 4) º) Verify that: (x) = (x 3). 4º) alculate the dimensions of B when the area of EFG is equal to 5 cm. Exercise XII: In the adjacent figure we have two rectangles B and EFG. x is a real number such that x > 7. 1º) Show that the area of the shaded region is: (x) = 3x 1 cm. º) Find the dimensions of the rectangle B when (x) = 180. Exercise XIII: 1º) alculate a when is a root of the polynomial: E (x) = x a + 3( x) (x 1). º) onsider the polynomial: P (x) = x + 3( x) (x 1) 8. Solve the equations: P (x) + 14 = 0 and P (x) = 0. x E 1 x + 1 3x - x + 10 x - 5 F J G B October - Math S.S-4 lgebraic expressions Grade 9 4/5

5 Exercise XIV: Given the expression: E(x) = (x - ) (3x + 1) (x ) (x + 10). 1º) Factorize E(x), and then determine its roots. º) In this question, x is a real number expressed in cm such that x >. onsider a rectangle B such that B = x - and B = 3 + 1, and a triangle MNP right at M, such that MN = (x ) and MP = x We designate by S the area of B and by S that of MNP. a) Express S and S in terms of x, and then show that: S S = E (x). b) alculate x so that that S = S. Exercise XV: Let B be a right triangle at such that : = x + 1 and B = x + 3 P (x) is a polynomial defined by P (x) = B. (x is a real number expressed in cm). 1º) Explain why x must be greater than 0.5 cm. º) Find the expression of P (x). 3º) a) For what value of x B is it isosceles at? b) alculate B for the obtained value. 4º) S(x) designates the area of the triangle B. a) Write the reduce expression of S(x), and then calculate S(x) when x =. b) educe the length of [H], the height-segment relative to [B]. Exercise XVI: EFG is a right triangle at E such that EF = 3x 3 and EG = 3x 1. x is a real number expressed in cm such that x > 1. 1º) I is the midpoint of [EG] and we know that EI = x + 1. Express IG in terms of x. º) Show that the area of the triangle FGI is given by U (x) = (3x 3)(x 1). 3 ) Find the value of x for which EFI and FGI have two equal areas October - Math S.S-4 lgebraic expressions Grade 9 5/5

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