MULTIPLYING ALGEBRAIC TERMS AND EXPRESSIONS

Problem 1 :

What is the power of X in the product of X × X.

a)  4    b)  2      c)  3     d)  None of these

Solution :

Product of X × X = X²

 The power of X = 2

So, option (b) is correct.

Problem 2 :

Z × (3 + Z) = ______

Solution :

= Z × (3 + Z)

= Z × 3 + Z × Z

= 3Z + Z²

Problem 3 :

Find the product, 4X × 3Y × 5.

Solution :

= 4X × 3Y × 5

= 60XY

Problem 4 :

Find the product, (3x + 2) (5)

Solution :

= (3x + 2) × (5)

= 15x + 10

Problem 5 :

4.1X × 3.4Y × 5.6Z find the product.

Solution :

= 4.1X × 3.4Y × 5.6Z

= 78.064 XYZ

Problem 6 :

Find the product of 4x × 0 × 24y

Solution :

= 4x × 0 × 24y

= 0

Always the product of any number and zero will be zero.

Problem 7 :

Solve the following (3x + 7) (2y + 1)

Solution :

= (3x + 7) (2y + 1)

= 3x (2y + 1) + 7(2y + 1)

= 6xy + 3x + 14y + 7

Problem 8 :

Find the product (3x + 2) (5 + 2x)

Solution :

= (3x + 2) (5 + 2x)

= 3x (5 + 2x) + 2 (5 + 2x)

= 15x + 6x² + 10 + 4x

= 6x² + 19x + 10

Problem 9 :

Find the product of

(3x + 2) and (5 + 2xy + x)

and write number of terms after multiplication.

Solution :

= (3x + 2) (5 + 2xy + x)

= 3x (5 + 2xy + x) + 2 (5 + 2xy + x)

= 15x + 6x²y + 3x² + 10 + 4xy + 2x

= 17x + 6x²y + 3x² + 10 + 4xy

There are 5 terms.

Problem 10 :

Write the product of variables in the expression

3x + 5yz + 7x²

Solution :

Variables = x, yz, x²

Product of variables = x × yz × x²

= x³yz

Problem 11 :

Find the product of (3x + 12y²) and (5 + 2xy + x)

Solution :

= (3x + 12y²) (5 + 2xy + x)

= 3x (5 + 2xy + x) + 12y² (5 + 2xy + x)

= 15x + 6x²y + 3x² + 60y² + 24xy³ + 12xy²

Problem 12 :

Z × (3 + 4Z) = ______

Solution :

= Z × (3 + 4Z)

= Z × 3 + Z × 4Z

= 3Z + 4Z²

Problem 13 :

Write a polynomial that represents the area of the shaded region.

Solution :

Area of rectangle = length ⋅ width

Length = 2x - 9

Width = x + 5

= (2x - 9)(x + 5)

= 2x(x) + 2x(5) - 9(x) - 9(5)

= 2x² + 10x - 9x - 45

= 2x² + x - 45

Problem 14 :

(2x + 3)(ax - 5) = 12x² + bx - 15

In the given equation, a and b are constants. If the equation is true for all values of x, what is the value of b ?

a)  6          b)  8        c)  10       d)  12

Solution :

(2x + 3)(ax - 5) = 12x² + bx - 15

2x(ax) + 2x(-5) + 3(ax) + 3(-5) = 12x² + bx - 15 

2ax² - 10x + 3ax - 15 = 12x² + bx - 15 

Equating the coefficient of x², we get

2a = 12

a = 12/2

a = 6

Equating the coefficient of x, we get

-10 + 3a = b

Applying the value of a, we get

-10 + 3(6) = b

-10 + 18 = b

b = 8

So, the value of b is 8, option b is correct.

Problem 15 :

The expression

5x⁵ + 6x⁴ - 8x³

can be rewritten as (x³ - hx²)(5x² + 10x), where h is a constant. What is the value of h ? 

Solution :

5x⁵ + 6x⁴ - 8x³ = (x³ - hx²)(5x² + 10x)

= 5x⁵ + 10x⁴ - 5hx⁴ - 10hx³

5x⁵ + 6x⁴ - 8x³ = 5x⁵ + (10 - 5h)x⁴ - 10hx³

10 - 5h = 6

10 - 6 = 5h

5h = 4

h = 4/5

So, the value of h is 4/5.

Problem 16 :

18x²  - 8 = 2(ax + b) (ax - b)

In the given equation, a and b are constants. If the equation is true for all values of x, which of the following could be the value of ab ?

a)  6      b)  9     c)  12      d)  36

Solution :

18x²  - 8 = 2(ax + b) (ax - b)

18x²  - 8 = 2[(ax)² - b²]

= 2[a²x² - b²]

= 2a²x² - 2b²

ab = 3(2) ==> 6

So, the value of a b is 6. 

Option a is correct.