PRACTICE PROBLEMS ON MULTIPLYING BINOMIALS

Expand and simplify :

1) (x + 3) (x + 7) 2)  (x + 5) (x - 4) 3) (x - 3)(x + 6) 4) (x + 2) (x - 2) 5)  (x - 8) (x + 3) 6)  (2x + 1) (3x + 4)

7)  (1 - 2x) (4x + 1) 8)  (4 - x) (2x + 3) 9)  (3x - 2) (1 + 2x) 10)  (5 - 3x)(5 + x) 11)  (7 - x)(4x + 1) 12)  (5x + 2) (5x + 2)

13)  A square photograph has sides of length x cm. It is surrounded by a wooden frame with the dimension shown. Show that the area of the rectangle formed by the outside of the frame is given by A = x² + 10x + 24 cm²

Detailed Solution

Problem 1 :

(x + 3) (x + 7)

Solution :

= (x + 3) (x + 7)

= x(x) + x(7) + 3(x) + 3(7)

= x² + 7x + 3x + 21

= x² + 10x + 21

So, the answer is x² + 10x + 21.

Problem 2 :

(x + 5) (x - 4)

Solution :

= (x + 5) (x - 4)

= x(x) + x(-4) + 5(x) + 5(-4)

= x² - 4x + 5x - 20

= x² + x - 20

So, the answer is x² + x - 20.

Problem 3 :

(x - 3)(x + 6)

Solution :

= (x - 3)(x + 6)

= x(x) + x(6) - 3(x) - 3(6)

= x² + 6x - 3x - 18

= x² + 3x - 18

So, the answer is x² + 3x - 18.

Problem 4 :

(x + 2) (x - 2)

Solution :

= (x + 2) (x - 2)

= x(x) + x(-2) + 2(x) + 2(-2)

= x² - 2x + 2x - 4

= x² - 4

So, the answer is x² - 4.

Problem 5 :

(x - 8) (x + 3)

Solution :

= (x - 8) (x + 3)

= x(x) + x(3) - 8(x) - 8(3)

= x² + 3x - 8x - 24

= x² - 5x - 24

So, the answer is x² - 5x - 24

Problem 5 :

(2x + 1) (3x + 4)

Solution :

= (2x + 1) (3x + 4)

= 2x(3x) + 2x(4) + 1(3x) + 1(4)

= 6x² + 8x + 3x + 4

= 6x² + 11x + 4

So, the answer is 6x² + 11x + 4.

Problem 6 :

(1 - 2x) (4x + 1)

Solution :

= (1 - 2x) (4x + 1)

= 1(4x) + 1(1) - 2x(4x) - 2x(1)

= 4x + 1 - 8x² - 2x

Combining the like terms, we get

= 2x + 1 - 8x² 

Arranging from greatest exponent to least exponent, we get

= - 8x² + 2x + 1

So, the answer is - 8x² + 2x + 1

Problem 8 :

(4 - x) (2x + 3)

Solution :

= (4 - x) (2x + 3)

= 4(2x) + 4(3) - x(2x) - x(3)

= 8x² + 12 - 2x² - 3x

Combining the like terms, we get

= 8x² - 2x² + 12 - 3x

=  6x² - 3x + 12

So, the answer is 6x² - 3x + 12.

Problem 9 :

(3x - 2) (1 + 2x)

Solution :

= (3x - 2) (1 + 2x)

= 3x(1) + 3x(2x) - 2(1) - 2(2x)

= 3x + 6x² - 2 - 4x

Combining the like terms, we get

= 6x² - 4x + 3x - 2

= 6x² - x - 2

So, the answer is 6x² - x - 2

Problem 10 :

(5 - 3x)(5 + x)

Solution :

= (5 - 3x)(5 + x)

= 5(5) + 5(x) - 3x(5) - 3x(x)

= 25 + 5x - 15x - 3x²

Combining the like terms, we get

= - 3x² - 15x + 5x + 25

= - 3x² - 10x + 25

So, the answer is - 3x² - 10x + 25.

Problem 11 :

(7 - x)(4x + 1)

Solution :

= (7 - x)(4x + 1)

= 7(4x) + 7(1) - x(4x) - x(1)

= 28x + 7 - 4x² - x

Combining the like terms, we get

= - 4x² + 28x - x + 7

= - 4x² + 27x + 7

So, the answer is - 4x² + 27x + 7.

Problem 12 :

(5x + 2) (5x + 2)

Solution :

= (5x + 2) (5x + 2)

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

= 25x² + 10x + 10x + 4

Combining the like terms, we get

= 25x² + 20x + 4

Problem 13 :

A square photograph has sides of length x cm. It is surrounded by a wooden frame with the dimension shown. Show that the area of the rectangle formed by the outside of the frame is given by A = x² + 10x + 24 cm²

Solution :

Area of small square = side x side

side length = x

Area of large rectangle = length x width

Length = x + 3 + 3 ==> x + 6

width = x + 2 + 2 ==> x + 4

Area of outside rectangle = (x + 6)(x + 4)

= x² + 4x + 6x + 6(4)

= x² + 10x + 24

So, it is proved.