Basic algebra 63

= 5 × 2 ×

(

2

5

) 2

×

(

5

2

) 3

since 2

1

2

=

5

2

=

5

1

×

2

1

×

2

5

×

2

5

×

5

2

×

5

2

×

5

2

=

1

1

×

1

1

×

1

1

×

1

1

×

1

1

×

5

1

×

5

1

by cancelling

= 5 × 5

= 25

Problem 10. Multiply 2a+ 3 bbya+b

Each term in the first expression is multipliedbya,then
each term in the first expression is multiplied byband
the two results are added. The usual layout is shown
below.
2 a+ 3 b
a+b
Multiplying byagives 2a^2 + 3 ab
Multiplying bybgives 2 ab+ 3 b^2

Adding gives^2 a

(^2) + 5 ab+ 3 b 2
Thus,( 2 a+ 3 b)(a+b)= 2 a^2 + 5 ab+ 3 b^2
Problem 11. Multiply 3x− 2 y^2 + 4 xyby
2 x− 5 y
3 x − 2 y^2 + 4 xy
Multiplying^2 x −^5 y
by 2x→ 6 x^2 − 4 xy^2 + 8 x^2 y
Multiplying
by− 5 y→− 20 xy^2 − 15 xy+ 10 y^3
Adding gives 6 x^2 − 24 xy^2 + 8 x^2 y− 15 xy+ 10 y^3
Thus,( 3 x− 2 y^2 + 4 xy)( 2 x− 5 y)
= 6 x^2 − 24 xy^2 + 8 x^2 y− 15 xy+ 10 y^3
Problem 12. Simplify 2x÷ 8 xy
2 x÷ 8 xymeans
2 x
8 xy
2 x
8 xy

2 ×x
8 ×x×y

1 × 1
4 × 1 ×y
by cancelling

1
4 y
Problem 13. Simplify
9 a^2 bc
3 ac
9 a^2 bc
3 ac

9 ×a×a×b×c
3 ×a×c
= 3 ×a×b
= 3 ab
Problem 14. Divide 2x^2 +x−3byx− 1
(i) 2x^2 +x−3iscalledthedividendandx−1the
divisor. The usual layoutis shown below withthe
dividend and divisorboth arranged in descending
powers of the symbols.
2 x+ 3
x− 1
)
2 x^2 +x− 3
2 x^2 − 2 x
3 x− 3
3 x− 3
..
(ii) Dividing the first term of the dividend by the
first term of the divisor, i.e.
2 x^2
x
gives 2x,which
is put above the first term of the dividend as
shown.
(iii) The divisor is then multiplied by 2x,i.e.
2 x(x− 1 )= 2 x^2 − 2 x, which is placedunder the
dividend as shown. Subtracting gives 3x−3.
(iv) The process is then repeated, i.e. the first term
of the divisor,x, is divided into 3x,giving+3,
which is placed above the dividend as shown.
(v) Then 3(x− 1 )= 3 x−3, which is placedunder
the3x−3. The remainder, on subtraction, is zero,
which completes the process.
Thus,( 2 x^2 +x− 3 )÷(x− 1 )=( 2 x+ 3 ).
(A check can be made on this answer by
multiplying( 2 x+ 3 )by(x− 1 ), which equals
2 x^2 +x−3.)
Problem 15. Simplify
x^3 +y^3
x+y