HOW TO DO ALGEBRA - SPECIAL PRODUCTS, EXAMPLES, EXERCISE, USING FORMULAS

Algebra Formulas before How To Do Algebra
if you have not already done so.
There we have listed out all the
Algebra Formulas that need to be
remembered.

Proofs of the first six Formulas

Proofs of the Last six Formulas

if you have not already done so.

Here we present Solved Examples
and Exercise problems
on application of those Formulas.

Example 1 of How To Do Algebra

Find the quotient (27x3 - 64)÷(9x2 + 12x + 16) without actual division.

Solution to Example 1 of How To Do Algebra :
Let P = (27x3 - 64)÷(9x2 + 12x + 16)
We know 27 = 3 x 3 x 3 = 33 ⇒27x3 = 33x3 = (3x)3
64 = 4 x 4 x 4 = 43
∴ Numerator of P = (27x3 - 64) = (3x)3 - 43
This looks like (a3 - b3) with (3x) in place of a and (4) in place of b
Denominator of P = (9x2 + 12x + 16) = (3x)2 + (3x)(4) + 42
This looks like (a2 + ab + b2) with (3x) in place of a and (4) in place of b
We have (a3 - b3) = (a - b)(a2 + ab + b2) (See Formula 5)
⇒ (a3 - b3)÷(a2 + ab + b2) = (a - b)
P is like the L.H.S. of this with (3x) in place of a and (4) in place of b
∴ P = (3x - 4) Ans.

Example 2 of How To Do Algebra

If 2x + 3y - 4z = 10 and 3xy - 6yz - 4zx = 15, find 4x2 + 9y2 + 16z2.

Solution to Example 2 of How To Do Algebra :
Let P = 4x2 + 9y2 + 16z2 = (2x)2 + (3y)2 + (4z)2
This looks like a2 + b2 + c2
with (2x) in place of a, (3y) in place of b and (4z) in place of c
Then the given data looks like a + b - c = 10 and (1⁄2 )(ab - bc - ca) = 15
We have (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca (See Formula 8)
Putting (-c) in place of c, we get
(a + b - c)2 = a2 + b2 + (-c)2 + 2ab + 2b(-c) + 2(-c)a (See Formula 8)
= a2 + b2 + c2 + 2ab - 2bc - 2ca
a2 + b2 + c2= (a + b - c)2 - 2(ab - bc - ca)
Applying this to P, we get
P = (2x)2 + (3y)2 + (4z)2
= (2x + 3y - 4z)2 - 2{(2x)(3y) - (3y)(4z) - (4z)(2x)}
= (2x + 3y - 4z)2 - 2 x 2{(x)(3y) - (3y)(2z) - (4z)(x)}
= (2x + 3y - 4z)2 - 4{(3xy) - (6yz) - (4zx)}
By data, 2x + 3y - 4z = 10 and 3xy - 6yz - 4zx = 15
Using these in P, we get
P = (10)2 - 4(15) = 100 - 60 = 40. Ans.

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Example 3 of How To Do Algebra

Solved Example 3 on How To Do Algebra :

Find the quotient
(x3 + 27y3 + 8z3 - 18xyz)÷(x + 3y + 2z)
without actual division.

Solution to Example 3 of How To Do Algebra :
We have a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca) [See Formula 9]
⇒ (a3 + b3 + c3 - 3abc)÷(a + b + c) = (a2 + b2 + c2 - ab - bc - ca).....(i)
Let P = (x3 + 27y3 + 8z3 - 18xyz)÷(x + 3y + 2z)
= {x3 + (3y)3 + (2z)3 - 3(x)(3y)(2z)}÷(x + 3y + 2z)
This looks like Equation (i)
with x in place of a, 3y in place of b and 2z in place of c
Applying Equation (i) here, we get
P = {x3 + (3y)3 + (2z)3 - 3(x)(3y)(2z)}÷(x + 3y + 2z)
= {(x)2 + (3y)2 + (2z)2 - (x)(3y) - (3y)(2z) - (2z)(x)}
= {x2 + 9y2 + 4z2 - 3xy - 6yz - 2zx} Ans.

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Example 4 of How To Do Algebra

Solved Example 4 on How To Do Algebra :

Find the products
(i) (x + 1)(x - 3) (ii) (2x + 5)(5x - 3) (iii) (x + 2)(x + 4)(x + 5)
using the Algebra Formulas.

Solution to Example 4 of How To Do Algebra :
(i) Let P = (x + 1)(x - 3)
We have (x + a)(x + b) = x2 + x(a + b) + ab [See Formula 10]
Comparing P with this Formula, a = 1 and b = -3
∴ P = (x + 1)(x - 3) = x2 + x{1 + (-3)} + {(1)(-3)}
= x2 + x{-2} + {(-3)} = x2 - 2x - 3 Ans.

(ii) Let P = (2x + 5)(5x - 3)
We have(ax + b)(cx + d) = acx2 + x(ad + bc) + bd
Comparing P with this Formula, a = 2, b = 5, c = 5 and d = -3
∴ P = (2x + 5)(5x - 3) = (2)(5)x2 + x{(2)(-3) + (5)(5)} + (5)(-3) = 10x2 +19x - 15. Ans.

Example 5 of How To Do Algebra

Solved Example 5 on How To Do Algebra :

If l + m + n = 0, prove that l3 + m3 + n3 = 3lmn

Solution to Example 5 of How To Do Algebra :
l + m + n = 0 ⇒ l + m = -n .........(i)
Cubing both sides, we get
(l + m)3 = (-n)3
l3 + m3 + 3lm(l + m) = -n3 [See Formula 6]
Using the value of (l + m) from (i), we get
l3 + m3 + 3lm(-n) = -n3
l3 + m3 - 3lmn = -n3
l3 + m3 + n3 = 3lmn (Proved.)

Exercise : How To Do Algebra

Solve the following problems on How To Do Algebra

1. Find the quotient (8x3 - 343)÷(2x - 7) without actual division.
2. If x + 2y + 3z = 12 and x2 + 4y2 + 9z2 = 44, find 2xy + 6yz + 3zx
3. Find the quotient
(l3 + 27m3 + 343n3 - 63lmn)÷(l2 + 9m2 + 49n2 - 3lm - 21mn - 7nl)
without actual division.
4. Find the products
(i) (x - 9)(x - 7) (ii) (5x + 2)(7x + 5) (iii) (x - 3)(x - 8)(x + 6)
using the Algebra Formulas.
5. If p + q = r, prove that p3 + q3 + 3pqr = r3
For Answers See at the bottom of the page.

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Answers to Exercise : How To Do Algebra

Answers to Problems on How To Do Algebra :

1. 4x2 + 14x + 49
2. 50
3. l + 3m + 7n
4. (i) x2 -16x + 63 (ii) 35x2 + 39x + 10 (iii) x3 - 5x2 - 42x + 144

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