Grade 9 / double distributivity 56 exercises (100% corrected)

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ABCDEFGHMxx51 R1R2R3R4abdc x22x4x×2xx×42×2x2×42x24x4x82x28x8 ABCDEFGHIabcd a−bc−da×c−a×d−b×cb×c 3x×x33x×x3x×3 3x2x33x×x3x×32×x2×33x29x2x63x211x6 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
abcda×ca×db×cb×c 3x12x43x×2x3x×41×2x1×46x212x2x46x214x4 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 following 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 following 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 expression : A = 5 x − 1 2 x − 3 + 5 x − 3 E.3606 Expand then reduce each of the fol-lowing 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
fx123456ABCDEFÉtape125710Étape2811131626Étape3-302515Étape4-2402680390Étape5(résultat)63056110450Sommedunombreetdesoncarré63056110420B6B1B Choisir un nombreMultiplier 2Ajouter4Soustraire 5Multiplier les deux nombres Choisir un nombreMultiplier 3Ajouter2Soustraire 5Multiplier les deux nombres Choisir un nombre Choisir un nombreSoustraire2Soustraire3Multiplier les deux nombresAjouter1Programme BProgramme AChoisir un nombrePrendrelecarrédunombreMultiplier par3Soustraire15Diviser par3Ajouter5 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
Choisir un nombreMultiplier 3Soustraire3Soustraire 1Multiplier les deux nombresMultiplier par 10Additionner les deux nombres Choisir un nombreAjouter3Soustraire3Multiplier les deux nombresAjouter9 Choisir un nombreAjouter3Soustraire 4Multiplier les deux nombresAdditionner les deux nombres xcmxcmABCDEFGH2cm1;5cmI xx7m4m 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
ABC5x+103x+64x+8 ABC5x+103x+64x+8 ABCH 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