Outside the high school program / Algebra 32 exercises corrected

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1. Area of resolution, definition E.1343 Here is the work of a student trying to solve the equation : (1+ x ) 2 =1+ x 1 a Verify that the value − 1 is a solution of this equa-tion. b What can be said about the student’s proposed solu-tion? 2 The diagram below represents the different steps per-formed by this student. Complete the ˇ labels ı by in-dicating the action performed by the student : 3 a For what values of x are defined the two expressions : (1 + x ) 2 ; 1 + x b For what values of x are defined the two expressions : (1 + x ) 2 1 + x ; 1 + x 1 + x c Explain where the error made by the student comes from. E.4389 We wish to solve the equation : ( E ): ( x − 2) x x +1 = 2 x 2 +1 x +1 1 What is the solving range of this equation? 2 For x = − 1 , establish the following equality: 2 x 2 + 1 x + 1 − ( x − 2) x x + 1 = x + 1 3 What can be said about the set of solutions of ( E ) ? E.4402 The diagram below shows the steps in-volved in solving an algebraic equation. Briefly describe each of these steps, indicating in the la-bels the mathematical action performed (factoring, expan-sion, subtraction. . . ) . E.4544 Consider the function f defined by the relation: f ( x ) = − 2 x + 3 1 Justify that the defining set of the function f is R + . 2 Establish that the function f is strictly increasing on R + . E.454 Solve the following equation : 2 x − 2 x − 1 = 3 x +3 2 x +1 2. Handling frames, inequalities E.1952 from Euclid’s method for ob-taining Pythagorean triplets a = m 2 − n 2 .... Show why m 2 − n 2 < 2 mn 3. Development E.6994 Give the reduced developed form of the following expressions : a 2 x + 1 x + 2 b x − 1 (2 x + 2 c − 2 − x 3 x + 1 d 2 x + 3 2 https://chingmath.fr chapExoCorrec/1343 sacados/1343 chapExoCorrec/4389 sacados/4389 chapExoCorrec/4402 sacados/4402 5(x2x3(x25x102x3x1527x103x177x3x17−107x3x77x−3x74x7x74 chapExoCorrec/4544 sacados/4544 chapExoCorrec/454 sacados/454 sacados/1952 a faire chapExoCorrec/6994 sacados/6994
4. Factoring E.4424 Factor the following expressions : a (3 x − 1)(2 x + 1) + (5 − x )(2 x + 1) b x (2 − x ) + (3 x + 1)(2 − x ) c ( x + 1)( x − 1) − (2 x + 3)( x − 1) E.4403 Solve the following equations : a (4 x + 3)(2 − 3 x ) + (2 − 6 x )(2 − 3 x ) = 0 b 2 · ( x − 2)(3 x + 2) = 3(2 x + 3)( x − 2) c ( x + 1)(3 x − 2) = ( x + 1) 2 E.9423 Factor the following expressions : a (3 x + 4)(2 x − 1) + 4(3 x + 4) b (2 x + 4)(3 − 3 x ) + (2 x + 4) c ( x + 1)(3 − 2 x ) + (3 − 2 x ) 2 E.9426 Consider equation : ( F ) : 3 − 3 x 2 x + 2 + 3 x − 2 2 x + 3 = 0 1 Establish the following equality: 3 − 3 x 2 x + 2 + 3 x − 2 2 x + 3 = 5 − x (2 x + 2)(2 x + 3) 2 Solve equation ( F ) . 5. Factoring by a common factor E.9420 Factor the following expressions : a ( x + 2)( x − 3) + (3 x + 6)(2 x − 1) b (3 x + 6)( x − 4) − (2 x − 8)( x − 2) c (2 x + 7)(27 x + 18) − (5 − x )(3 x + 2) E.9422 Factor the following expressions : a (3 x + 2)(5 − x ) + (6 x + 4) b (5 − 2 x )( x + 1) − 3 x (2 x − 5) c (5 − 2 x ) + (4 x − 10)(7 − 2 x ) E.9418 Factor the following expressions : a (3 x + 2)(4 x + 2) − (5 x + 1)(2 x + 1) b (2 x − 1)(9 x + 3) + (3 x + 1) 2 E.4613 Reminders: The opposite of the expression 3 x − 4 can be expressed with the following two forms : − 3 x +4 ; 4 − 3 x The opposite of the product a × b can be expressed with the two forms : ( − a ) × b ; a × ( − b ) Attention, the product ( − a ) × ( − b ) is égal to the product a × b . Applications: In the expression (3 x − 4)(5 x +1)+(4 − 3 x )(2 − 2 x ) , the common factor is 3 x − 4 . We have the following transformations : (2 − x )( x − 4) = − ( x − 2) ( x − 4) = ( x − 2)(4 − x ) Factor the following expressions : a (3 x − 4)(5 x + 1) + (4 − 3 x )(2 − 2 x ) b ( x − 2)(2 x + 3) + (2 − x )( x − 4) E.6995 Reminder : when the coefficients of a polynomial are mul-tiples of the same number, a factorization is possible. Example: 2 x + 2 = 2 · x + 1 6 x − 3 = 3 · 2 x − 1 Factor the following expressions : a x + 1 2 x − 1 + 2 x + 2 3 x + 2 b 6 x − 3 x + 1 + 2 x − 1 x + 2 c 9 x − 3 (3 x + 1) − x − 2 3 x − 1 d 2 x + 2 2 + x + 1 E.7219 Factor the following expressions : a x + 3 2 x − 1 − 3 x + 1 x + 3 b 6 − 3 x x + 2 − 2 − x 2 6. Factorisation by a common factor and equation https://chingmath.fr chapExoCorrec/4424 sacados/4424 chapExoCorrec/4403 sacados/4403 chapExoCorrec/9423 sacados/9423 chapExoCorrec/9426 sacados/9426 chapExoCorrec/9420 sacados/9420 chapExoCorrec/9422 sacados/9422 chapExoCorrec/9418 sacados/9418 chapExoCorrec/4613 sacados/4613 chapExoCorrec/6995 sacados/6995 chapExoCorrec/7219 sacados/7219
E.9414 Solve the following equations : a (4 x + 2)(3 x − 1) + x (6 x + 3) = 0 b (3 x + 9) 2 = 2(2 x + 6)(3 x − 2) E.2858 Solve the following equations : a (3 x + 1)(1 − 3 x ) + (6 x + 2)(3 x − 1) = 0 E.2113 Solve the following equations using the method of your choice : a (2 x − 3)(4 x − 2) + (3 − 6 x )( x − 3) = 0 b (2 − 6 x )(3 x + 3) = ( − 9 x + 3)(2 x + 1) E.9416 Solve the following equations : a (5 x + 2)(4 x − 3) = (2 x − 1)(3 − 4 x ) b (2 − 3 x )(2 x − 4) + (10 − 5 x )(3 x − 1) = 0 E.9419 Solve the following equation : (5 x + 1)(3 x − 1) − 2( x + 1)(1 − 3 x ) = 0 E.4566 Solve the following equations : a ( x + 2)(3 − x ) + 2( x − 3)(2 x − 5) = 0 b (6 − 2 x )(3 x + 2) = (3 x − 9)( x + 2) c (9 − 3 x )(5 + 2 x ) + ( x − 3)(5 + x ) = 0 E.4425 Factor the following expressions : a ( x − 1)(2 x + 1) − (2 x − 2)(5 − 2 x ) b 3(4 + 2 x ) − (3 + x )(10 + 5 x ) c (2 − x )(3 x − 4) + 2 − 3 2 x (2 x + 3) 7. Factorization: remarkable identity E.9421 Factor the following expressions : a (2 x + 1) 2 − 4(2 − 3 x ) 2 b 18 x 2 − 24 x + 8 + (3 x − 2)(2 − x ) E.4528 Factor the following expressions : a − 5(3 x + 2) − 5 x (2 x + 1) 8. Factoring by common factor and remarkable identity E.4551 Factor the following expressions : a (3 x + 2)(5 x − 1) − (2 − 10 x )(2 − 4 x ) b ( x + 3)(3 x + 6) + (4 x + 8) 2 c (2 x + 3)(5 x − 4) − (2 x − 2)(6 − x ) d ( x − 3)( − 3 x − 3) − (2 x + 2)(3 − 3 x ) E.9425 Solve the following equations : a (1 − 5 x )(2 x + 1) = (1 − 3 x )(3 x + 2) b (6 x + 1)(3 x + 1) + (2 x + 1)(2 − 9 x ) = 0 c ( − 5 x − 4)(2 x + 1) = ( − 4 x − 3)(3 x + 2) E.4547 Solve the following equations : a (4 x + 2)(3 x − 1) + x (6 x + 3) = 0 b ( x + 3) 2 = 2(2 x + 6)(3 x − 2) c ( − 5 x − 4)(2 x + 1) = ( − 4 x − 3)(3 x + 2) d (1 − 5 x )(2 x + 1) = (1 − 3 x )(3 x + 2) 9. Equations E.1940 Consider two real numbers x and y ver-ifying the following frames : − 4 x − 1 ; 2 y 5 1 Give the frames of the following numbers : a 3 × (1 − x ) + 1 b 1 − 3 − x 4 2 Give the frames of the following numbers : a x 2 b x × y c 2 x − 3 y https://chingmath.fr chapExoCorrec/9414 sacados/9414 chapExoCorrec/2858 sacados/2858 chapExoCorrec/2113 sacados/2113 chapExoCorrec/9416 sacados/9416 chapExoCorrec/9419 sacados/9419 chapExoCorrec/4566 sacados/4566 chapExoCorrec/4425 sacados/4425 chapExoCorrec/9421 sacados/9421 chapExoCorrec/4528 sacados/4528 chapExoCorrec/4551 sacados/4551 chapExoCorrec/9425 sacados/9425 chapExoCorrec/4547 sacados/4547 chapExoCorrec/1940 sacados/1940
10. Square roots E.296 1 Let a and b be two numbers such that a 0 and b 0 . Compare numbers : a + b ; a + b 2 What conditions on a and b must there be for the inequal-ity to be strict? 11. Euclidean division E.2299 Here are two Euclidean divisions of poly-nomials : 2 x 3 +2 x 2 − 16 x − 12 2 x 3 +6 x 2 − 4 x 2 − 16 x − 4 x 2 − 12 x − 4 X − 12 − 4 X − 12 0 x + 3 2 x 2 − 4 x − 4 3 x 3 − 5 x 2 + x − 7 3 x 3 − 6 x 2 x 2 + x x 2 − 2 x 3 X − 7 3 X − 6 − 1 x − 2 3 x 2 + x + 3 They produce the following multiplicative entries : 2 x 3 + 2 x 2 − 16 x − 12 = ( x + 3)(2 x 2 − 4 x − 4) This equality yields the three roots of this second-degree polynomial. 3 x 3 − 5 x 2 + x − 7 = ( x − 2)(3 x 2 + x + 3) − 1 This equality yields the following identity called ˇ simple element decomposition of a rational fraction ı : 3 x 3 − 5 x 2 + x − 7 x − 2 = 3 x 2 + x + 3 − 1 x − 2 Using Euclidean division of polynomials, answer the following questions : 1 Noting that − 3 is a root of x 3 + 3 x 2 − 2 x + 3 , determine that this polynomial admits no other roots. 2 Show that : 2 x 3 − 5 x 2 + x − 4 2 x + 1 = x 2 − 3 x + 2 − − 6 2 x + 1 12. Unclassified financial years E.7332 Consider the two functions f and g de-fined on R + by the relations : f ( x ) = 2 · x − 1 · x ; g ( x ) = 3 · x 2 − 9 · x + 6 Using the calculator, conjecture the relative position on R + of the curves C f and C g representative of the functions f and g respectively. Calculator results rounded to the nearest hundredth will be used. E.2681 Say whether the following assertions are true or false : 1 For a quotient to be zero, its numerator must be zero. 2 For a quotient to be zero, it is sufficient for its numerator to be zero. https://chingmath.fr chapExoCorrec/296 sacados/296 chapExoCorrec/2299 sacados/2299 chapExoCorrec/7332 sacados/7332 sacados/2681