Exploring Algebraic Identities | EOT

Question 13

Find the value of

(i) x3+y3−12xy+64x^3 + y^3 - 12xy + 64, when x+y=−4x + y = -4

(ii) x3−8y3−36xy−216x^3 - 8y^3 - 36xy - 216, when x=2y+6x = 2y + 6

Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question

To find the values of these algebraic expressions, we use the algebraic expansion of the cube of a binomial:

  • (a+b)3=a3+b3+3ab(a+b)(a + b)^3 = a^3 + b^3 + 3ab(a + b)
  • (a−b)3=a3−b3−3ab(a−b)(a - b)^3 = a^3 - b^3 - 3ab(a - b)

By cubing both sides of the given linear equations and substituting the known values, the required cubic expressions can be directly evaluated.

(i) Find the value of x3+y3−12xy+64x^3 + y^3 - 12xy + 64, when x+y=−4x + y = -4

Step 1 · Cube both sides of the given equation

Given x+y=−4x + y = -4

Cubing both sides (x+y)3=(−4)3(x + y)^3 = (-4)^3

x3+y3+3xy(x+y)=−64x^3 + y^3 + 3xy(x + y) = -64

Substitute x+y=−4x + y = -4

x3+y3+3xy(−4)=−64x3+y3−12xy=−64x3+y3−12xy+64=0\begin{aligned} x^3 + y^3 + 3xy(-4) &= -64 \\[0.6em] x^3 + y^3 - 12xy &= -64 \\[0.6em] x^3 + y^3 - 12xy + 64 &= 0 \end{aligned}
Answer

(i) 00

(ii) Find the value of x3−8y3−36xy−216x^3 - 8y^3 - 36xy - 216, when x=2y+6x = 2y + 6

Step 1 · Rearrange and cube both sides

Given x=2y+6  ⟹  x−2y=6x = 2y + 6 \implies x - 2y = 6

Cubing both sides (x−2y)3=63(x - 2y)^3 = 6^3

Using (a−b)3=a3−b3−3ab(a−b)(a - b)^3 = a^3 - b^3 - 3ab(a - b) x3−(2y)3−3(x)(2y)(x−2y)=216x^3 - (2y)^3 - 3(x)(2y)(x - 2y) = 216

x3−8y3−6xy(x−2y)=216x^3 - 8y^3 - 6xy(x - 2y) = 216

Substitute x−2y=6x - 2y = 6

x3−8y3−6xy(6)=216x3−8y3−36xy=216x3−8y3−36xy−216=0\begin{aligned} x^3 - 8y^3 - 6xy(6) &= 216 \\[0.6em] x^3 - 8y^3 - 36xy &= 216 \\[0.6em] x^3 - 8y^3 - 36xy - 216 &= 0 \end{aligned}
Answer

(ii) 00

Common Mistakes
  • Sign errors during cubing: Forgetting that (−4)3=−64(-4)^3 = -64 (a negative number cubed remains negative), or mishandling signs in the expansion of (x−2y)3(x - 2y)^3.
  • Incorrect binomial cube formula: Confusing (a−b)3=a3−b3−3ab(a−b)(a - b)^3 = a^3 - b^3 - 3ab(a - b) with a3−b3+3ab(a−b)a^3 - b^3 + 3ab(a - b).
  • Forgetting to substitute the inner term: Omitting the step where (x+y)(x + y) or (x−2y)(x - 2y) inside 3ab(… )3ab(\dots) is replaced by its given numerical value.

More questions in EOT

Q1

Use suitable identities to find the following products:

(i) (−3x+4)2(-3x + 4)^2

(ii) (2s+7)(2s−7)(2s + 7)(2s - 7)

(iii) (p2+12)(p2−12)\left(p^2 + \dfrac{1}{2}\right)\left(p^2 - \dfrac{1}{2}\right)

(iv) (2n+7)(2n−7)(2n + 7)(2n - 7)

(v) (s−2t)(s2+2st+4t2)(s - 2t)(s^2 + 2st + 4t^2)

(vi) (12r−4r)2\left(\dfrac{1}{2r} - 4r\right)^2

(vii) (−3m+4k−l)2(-3m + 4k - l)^2

(viii) (x−13y)3\left(x - \dfrac{1}{3}y\right)^3

(ix) (72k−23m)3\left(\dfrac{7}{2}k - \dfrac{2}{3}m\right)^3

Q2

Find the values using suitable identities:

(i) 17×2117 \times 21

(ii) 104×96104 \times 96

(iii) 24×1624 \times 16

(iv) 1473147^3

(v) 1993199^3

(vi) 1273127^3

(vii) (−107)3(-107)^3

(viii) (−299)3(-299)^3

Q3

Factor the following algebraic expressions:

(i) 4y2+1+116y24y^2 + 1 + \dfrac{1}{16y^2}

(ii) 9m2−125n29m^2 - \dfrac{1}{25n^2}

(iii) 27b3−164b327b^3 - \dfrac{1}{64b^3}

(iv) x2+5x6+16x^2 + \dfrac{5x}{6} + \dfrac{1}{6}

(v) 27u3−1125−27u25+9u2527u^3 - \dfrac{1}{125} - \dfrac{27u^2}{5} + \dfrac{9u}{25}

(vi) 64y3+1125z364y^3 + \dfrac{1}{125}z^3

(vii) p3+27q3+r3−9pqrp^3 + 27q^3 + r^3 - 9pqr

(viii) 9m2−12m+49m^2 - 12m + 4

(ix) 9x3−83y3+z33+6xyz9x^3 - \dfrac{8}{3}y^3 + \dfrac{z^3}{3} + 6xyz

(x) 4x2+9y2+36z2+12xz+36yz+24xy4x^2 + 9y^2 + 36z^2 + 12xz + 36yz + 24xy

(xi) 27u3−1216−9u22+u427u^3 - \dfrac{1}{216} - \dfrac{9u^2}{2} + \dfrac{u}{4}

Q4

Simplify the following:

(i) 4x2+4x+14x2−1\dfrac{4x^2 + 4x + 1}{4x^2 - 1}

(ii) 9(3a3−24b3)9a2−36b2\dfrac{9(3a^3 - 24b^3)}{9a^2 - 36b^2}

(iii) s3+125t3s2−2st−35t2\dfrac{s^3 + 125t^3}{s^2 - 2st - 35t^2}

Note: Assume that the denominators are not equal to 0.

Q5

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.

(i) 25a2−30ab+9b225a^2 - 30ab + 9b^2

(ii) 36s2−49t236s^2 - 49t^2

Q6

Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units.

(i) 6a2−24b26a^2 - 24b^2

(ii) 3ps2−15ps+12p3ps^2 - 15ps + 12p

Q7

The village playground is shaped as a square of side 40 metres. A path of width ss metres is created around the playground for people to walk. Find an expression for the area of the path in terms of ss.

Q8

If a number plus its reciprocal equals 103\dfrac{10}{3}, find the number.

Q9

A rectangular pool has area 2x2+7x+32x^2 + 7x + 3 square hastas. If its width is 2x+12x + 1 hastas, find its length. Hasta was a unit used to measure length.

Q10

If both x−2x - 2 and x−12x - \dfrac{1}{2} are factors of px2+5x+rpx^2 + 5x + r, show that p=rp = r.

Q11

If a+b+c=5a + b + c = 5 and ab+bc+ca=10ab + bc + ca = 10, then prove that a3+b3+c3−3abc=−25a^3 + b^3 + c^3 - 3abc = -25.

Q12

By factoring the expression, check that n3−nn^3 - n is always divisible by 6 for all natural numbers nn. Give reasons.

Q13

Find the value of

(i) x3+y3−12xy+64x^3 + y^3 - 12xy + 64, when x+y=−4x + y = -4

(ii) x3−8y3−36xy−216x^3 - 8y^3 - 36xy - 216, when x=2y+6x = 2y + 6

← Back to Exploring Algebraic Identities