Grade 9 / double distributivity 56 exercises corrected

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1. Introduction to double distributivity E.11105 Consider the figure below com-posed of rectangles : 1 Express the area of rectangle ABCD as a product. 2 Express the area of rectangle ABCD as a sum. E.7897 Consider a rectangle R divided into four rectangles R 1 , R 2 , R 3 , and R 4 : The dimensions are shown directly on the figure. 1 a Give the length and width of rectangle R . b Give an expression for the area A R of rectangle R . 2 a Express the areas A 1 , A 2 , A 3 , A 4 of the rectangles R 1 , R 2 , R 3 , R 4 respectively. b Give an expression for the area A R of rectangle R . We have just established the following property: Proposition: (double distributivity of multiplica-tion over addition) For all numbers a , b , c , d , we have the following identity: a + b c + d = a × c + a × d + b × c + b × d Example / Note: This expansion is often represented using the following ar-rows : E.1926 Consider the rectangle ABCD be-low. The lines ( EG ) and ( FH ) divide this rectangle into four rectangles. 1 Express the length and width of rectangle AEIH in terms of the measurements a , b , c , d , give an expression for the area of rectangle 2 By determining the areas of the four rectangles ABCD , ABCG , CDHF , GIFC , deduce another expression for the area of rectangle AEIH . Note: we have just established another identity of double distributivity: E.1924 Consider the following two expres-sions : A = (3 x + 2)( x + 3) ; B = 3 x 2 + 11 x + 6 1 a For each row of the following table, evaluate the two literal expressions. x (3 x + 2)( x + 3) 3 x 2 + 11 x + 6 0 2 − 1 b Make a conjecture about these two expressions. 2 Using simple distributivity, we can distribute the factor 3 x +2 over each term of the sum x +3 . We obtain the following identity: Let C be the expression : C = 3 x +2 × x + 3 x +2 × 3 a Expand, then reduce the expression C . b Justify the equality of expressions A and B are equal. Conclusion: we have just established the equality: https://chingmath.fr chapExoCorrec/11105 sacados/11105 ABCDEFGHMxx51 chapExoCorrec/7897 sacados/7897 R1R2R3R4abdc x22x4x×2xx×42×2x2×42x24x4x82x28x8 chapExoCorrec/1926 sacados/1926 ABCDEFGHIabcd a−bc−da×c−a×d−b×cb×c chapExoCorrec/1924 sacados/1924 3x×x33x×x3x×3 3x2x33x×x3x×32×x2×33x29x2x63x211x6
2. Double distributivity and positive coefficient E. 4599 Definition: " double distributivity "is the action of ex-panding two factors involving sums/and subtractions. Proposition: Let a , b , c , d be four numbers. We have the identity: Expand, then reduce the following expressions : a (3 x + 1)(3 + 4 x ) b (3 x + 2)(5 x + 4) c (2 x +5)(3+2 x ) d (2 x + 1)(3 x + 2) E. 1077 Expand, then reduce the following expressions : a (3 x + 1)(2 x + 4) b (2 x + 1)(3 x + 1) c (2 + x )( x + 2) d (2 x + 1)(7 + 4 x ) Commented example: E. 7938 Expand, then reduce the following expressions : a 2 x + 1) x + 3 b x + 4) x + 5 c x + 1) 2 x + 4 d 1 + 3 x ) 2 x + 1 E. 11501 Expand, then collapse the follow-ing expressions : a (3 x + 2)(2 x + 7) b ( x + 5)(4 x + 1) E.11490 Expand, then reduce the following expressions : E.8977 Expand, then collapse the following expressions : a ( x + 4) 2 b (2 x + 1) 2 3. Double distributivity E. 11401 Expand, then reduce the following expressions : E. 11402 Expand, then reduce the following expressions : E. 11421 Expand, then collapse the follow-ing expressions : a x + 5 2 x − 1 b 2 − 3 x 2 x + 1 E. 11422 Expand, then reduce the following expressions : E.11424 Expand, then reduce the following expressions : E.11423 Expand, then reduce the following expressions : E. 8994 Expand, then collapse the following expressions : E. 4623 Expand, then collapse the following expressions : a (5 − 2 x )( − 1+ x ) b ( − 3 x + 2)( x − 2) E.1079 Expand, then collapse the following expressions : a (2 x − 1)( − x − 3) b (5 x − 2)( − 3 − x ) E.1947 Expand, then reduce the following expressions : E.8963 Expand, then reduce the following expressions : a 2( − x − 2)(2 x − 6) b − ( x − 3)(7 − 2 x ) E.8966 Expand, then reduce the following expressions : a 2(5 x − 2)( x + 1) b − ( x + 1)( x + 1) E.1083 Copy the equations below, filling in the blanks with the missing expressions : a ( − 3 x + 1)( − x − 5) = : : : : : : x 2 + 14 x : : : : : : b (2 x + 5)( − 5 x + 2) = − 10 x 2 : : : : : : x + 10 c (3 x − 4)( − 2 x + 1) = − 6 x 2 : : : : : : x − 4 E.3605 Expand and reduce each of the fol-lowing expressions : a x + 1 2 x + 1 b 3 x + 1 2 x + 2 https://chingmath.fr chapExoCorrec/4599 sacados/4599 abcda×ca×db×cb×c chapExoCorrec/1077 sacados/1077 3x12x43x×2x3x×41×2x1×46x212x2x46x214x4 chapExoCorrec/7938 sacados/7938 chapExoCorrec/11501 sacados/11501 chapExoCorrec/11490 sacados/11490 chapExoCorrec/8977 sacados/8977 chapExoCorrec/11401 sacados/11401 chapExoCorrec/11402 sacados/11402 chapExoCorrec/11421 sacados/11421 chapExoCorrec/11422 sacados/11422 chapExoCorrec/11424 sacados/11424 chapExoCorrec/11423 sacados/11423 chapExoCorrec/8994 sacados/8994 chapExoCorrec/4623 sacados/4623 chapExoCorrec/1079 sacados/1079 chapExoCorrec/1947 sacados/1947 chapExoCorrec/8963 sacados/8963 chapExoCorrec/8966 sacados/8966 chapExoCorrec/1083 sacados/1083 chapExoCorrec/3605 sacados/3605
E.9201 Give the expanded and reduced forms of the following expressions : a 2 + x 3 x − 1 b 5 x − 1 x + 2 E.9198 Expand and reduce each of the fol-lowing expressions : a 2 x + 1 5 − 2 x b 3 x − 2 1 − x E.10809 Expand and simplify the following expressions : a 2 x + 1 x − 4 b 4 x − 1 5 − 2 x E.3607 Copy and complete the following equations correctly: a (3 x + 2)( : : : x + 1) = 15 x 2 + : : : x + : : : b ( x + 1)( x − : : : ) = : : : x 2 − x − : : : c (2 x + : : : )(1 + : : : x ) = − 4 x 2 + 4 x + : : : d (3 x + 1)( : : : x + : : : ) = 9 x 2 + : : : x + 1 E.2205 Copy and complete the following equations correctly: a (2 x + 3)(3 x + 1) = : : : x 2 : : : x : : : b (3 x + : : : )(2 : : : x ) = : : : x 2 + 7 x + 2 d ( : : : − x )( − 2 : : : x ) = x 2 − x − 6 e (3 x + : : : )( : : : x + : : : ) = 6 x 2 + 8 x + : : : E.9393 Expand and reduce each of the fol-lowing expressions : a − x + 1 2 x − 3 b 2 1 − x 2 − x 4. Double distribution and priority of operations E.703 Give the expanded and reduced form of the following expressions : a 2 × (2 x − 3)(0.5 x + 1) b (2 x + 1) 3 + (2 − x ) × 2 E. 10810 Expand and reduce the expres-sion : A = 5 x − 1 2 x − 3 + 5 x − 3 E. 3606 Expand then reduce each of the following expressions : a 3 x − 1 + x + 1 2 x + 1 b 2 − x 1 + x − 3 5 − 2 x E.8962 Expand and simplify the following expressions : a (5 − 2 x )(2 x + 4) + 3(5 − 3 x ) b (2 x + 3)( x − 4) + x (2 x + 4) E.11426 Expand and reduce the following expressions : a (5 x + 2)(3 x + 1) − ( x + 1) × 2 b ( x + 2)(3 x + 2) − (2 x + 3) × 4 E.8964 Determine the expanded and re-duced forms of each of the following expressions : E.9199 Expand and reduce each of the fol-lowing expressions : a x + 2 x + 1 + x + 3 2 x − 1 b 5 x − 1 x + 4 − 3 x + 2 E.10743 Expand then reduce each of the following expressions : a 3 x x − 1 − x − 2 2 x − 4 b (5 x + 1)(3 − x ) − 3(1 − x ) E.1078 Expand and reduce each of the fol-lowing literal expressions : b ( − x + 5)(3 x + 1) − ( − x + 2)(3 x + 4) E.11425 Expand and simplify the following expressions : a 3(5 x + 1)(2 − x ) − (5 − 2 x )(5 − 2 x ) b 2(2 − x )(3 x − 1) − 2 x (4 − 2 x ) E.8965 Expand and reduce the following expressions : a − (2 x + 1)(3 − x ) + 2( x − 1) b − (2 x − 1)( x +1) − (3 x +1)(2 − x ) 5. Double distributivity and problem https://chingmath.fr chapExoCorrec/9201 sacados/9201 chapExoCorrec/9198 sacados/9198 chapExoCorrec/10809 sacados/10809 chapExoCorrec/3607 sacados/3607 chapExoCorrec/2205 sacados/2205 chapExoCorrec/9393 sacados/9393 chapExoCorrec/703 sacados/703 chapExoCorrec/10810 sacados/10810 chapExoCorrec/3606 sacados/3606 chapExoCorrec/8962 sacados/8962 chapExoCorrec/11426 sacados/11426 chapExoCorrec/8964 sacados/8964 chapExoCorrec/9199 sacados/9199 chapExoCorrec/10743 sacados/10743 chapExoCorrec/1078 sacados/1078 chapExoCorrec/11425 sacados/11425 chapExoCorrec/8965 sacados/8965
E.8940 The following calculation program is given : Step 1 Choose a starting number Step 2 Add 6 to the starting number Step 3 Retract 5 to the starting number Step 4 Multiply the results of steps 2 and 3 Step 5 Add 30 to this product Step 6 Give the result 1 a Show that if the number chosen is 4 , the result is 20 . b What is the result when we apply this calculation pro-gram to the number − 3 ? 2 Zoe thinks that with a starting number chosen, the result is equal to the sum of that number and its square. a Check that she is right when the number chosen at the start is 4 , and also when − 3 is chosen. b Ishmael decides to use a spreadsheet to check Zoe’s assertion on some examples. He wrote formulas in B2 and B3 to automatically run steps 2 and 3 of the calculation program. What right-copy formula did he write in cell B4 to per-form step 4 ? c Zoe observes the results, then confirms that for any number x chosen, the result of the calculation program is x 2 + x Prove her answer. E.1948 1 Expand and reduce the following expression : A = ( x + 1)(2 x − 1) − x 2 Derive the value of the following calculation: B = 1 001 × 1 999 − 1 000 E. 7921 Consider the calculation program shown on the right : 1 By choosing the number 3 , show that as output the calculation program will return the number 12 . 2 Noting x the starting number, show that the number returned by the calculation program is expressed as : 2 x 2 +3 x − 20 E.11491 Consider the calculation program shown on the right : 1 By choosing the number 2 as the input value, de-termine the output value of this calculation pro-gram. 2 Let x be the input num-ber chosen for this calcu-lation program : a Give the expression for the output number as a function of x . b Establish that this expression can be expanded and reduced as follows : 3 x 2 + x − 10 E. 9030 Consider the expression E defined by: E = ( x − 5)(2 x + 1) Complete the diagram below, using instructions in French, so that the computation program constructed matches the expression E : E.11545 Consider the following two calcula-tion programs : 1 Show that when the chosen number is 4 , the result ob-tained with program A is 16 . 2 Show that when the chosen number is − 2 , the result ob-tained with program A is 4 . 3 Justify that the following statement is true : " Program A always gives the square of the chosen num-ber. " 4 When the chosen number is 10 , what result is obtained with program B ? 5 There is exactly one number for which programs A and B give identical results. What is this number? https://chingmath.fr chapExoCorrec/8940 sacados/8940 fx123456ABCDEFÉtape125710Étape2811131626Étape3-302515Étape4-2402680390Étape5(résultat)63056110450Sommedunombreetdesoncarré63056110420B6B1B chapExoCorrec/1948 sacados/1948 chapExoCorrec/7921 sacados/7921 Choisir un nombreMultiplier 2Ajouter4Soustraire 5Multiplier les deux nombres chapExoCorrec/11491 sacados/11491 Choisir un nombreMultiplier 3Ajouter2Soustraire 5Multiplier les deux nombres chapExoCorrec/9030 sacados/9030 Choisir un nombre chapExoCorrec/11545 sacados/11545 Choisir un nombreSoustraire2Soustraire3Multiplier les deux nombresAjouter1Programme BProgramme AChoisir un nombrePrendrelecarrédunombreMultiplier par3Soustraire15Diviser par3Ajouter5
E.7922 We consider the calculation program represented by the diagram below : Show that whatever number x is chosen as input, the out-put number returned by the computation program is always a strictly positive number. E.9008 The following calculation program is considered : 1 Give the expanded, then simplified expression associated with this calculation program. 2 Propose an equivalent but simplified calculus program. E.7937 We consider the calculation program represented by the diagram below : Determine the two possible numbers to choose as inputs to this calculation program so that the output number has the value 13 . E.2206 Consider the expression : A = (2 x + 1)( x + 2) − 2 x ( x − 1) 1 Expand and reduce the expression A . 2 Without using the calculator, deduce from the previous question the value of the following calculation: B = 20 001 × 10 002 − 20 000 × 9 999 6. Double distributivity, geometric figures and problems E.80 In the figure below, ABEG is a rect-angle and ABCD is a square. We are interested in rectangle AHFG . 1 a Find the area of rectangle AHFG when x =4 . b Consider the literal expression P defined by: P = x 2 + 0 ; 5 x − 3 Compute the expression P for x =4 . 2 Show that the area of rectangle AHFG can be expressed in terms of x using the algebraic expression P . E.6343 For this question, show the ap-proach used on the copy. Any trace of research will be taken into account during the evaluation even if the work is not completely completed. Juliette has a rectangular plot of land 7 m × 4 m on which she wants to build a pool. She wants to surround her pool with a driveway of the same width all around the pool. This situation is shown in the diagram below : Which of the following expressions represents the area of the pool: a x 2 − 11 x +28 b x 2 − 11 x − 28 c − x 2 +11 x +28 d 4 x 2 − 22 x +28 e 4 x 2 − 22 x − 28 f − 4 x 2 +22 x +28 https://chingmath.fr chapExoCorrec/7922 sacados/7922 Choisir un nombreMultiplier 3Soustraire3Soustraire 1Multiplier les deux nombresMultiplier par 10Additionner les deux nombres chapExoCorrec/9008 sacados/9008 Choisir un nombreAjouter3Soustraire3Multiplier les deux nombresAjouter9 chapExoCorrec/7937 sacados/7937 Choisir un nombreAjouter3Soustraire 4Multiplier les deux nombresAdditionner les deux nombres chapExoCorrec/2206 sacados/2206 chapExoCorrec/80 sacados/80 xcmxcmABCDEFGH2cm1;5cmI chapExoCorrec/6343 sacados/6343 xx7m4m
E.1949 Show that the triangle ABC is right-angled at C regardless of the value of ˇ x ı : E.7936 Show that the triangle ABC is right-angled at C regardless of the value of ˇ x ı : E.8758 The triangle ABC opposite is such that : AC = 13 ; AB = 14 ; BC = 15 We note H the foot of the height from the vertex C . Determine the measure of the segment [ CH ] . https://chingmath.fr chapExoCorrec/1949 sacados/1949 ABC5x+103x+64x+8 chapExoCorrec/7936 sacados/7936 ABC5x+103x+64x+8 chapExoCorrec/8758 sacados/8758 ABCH