FACTORING LINEAR EXPRESSIONS WORKSHEET

Question 1 : 

Factor :

4x + 8

Question 2 : 

Factor :

16a + 64b - 4c

Question 3 : 

Factor :

-25x + 15

Question 4 : 

Factor :

1.5y - 9

Question 5 : 

Factor : 

4w - 8x + 16y - 32z

Question 6 : 

Explain why 8x + 3y cannot be factorised.

Question 7 :

5k² - 40k + 10

Question 8 :

-x² + 16

Question 9 :

5p⁶ + 20p⁴ - 50p³

Question 10 :

The area of the rectangle is 12x² - 21x - 6, the width is 3x - 6, what is the length of the rectangle ?

Question 11 :

The volume of the rectangular prism is 6c³ - 22c² - 8c. Find the length and width of this prism if the height is 2c.

Question 12 :

Write the expression for the length of a rectangle with width y - 9 and area y² - 14y + 45

Detailed Answer Key

1. Answer : 

4x + 8

Find the largest common divisors for 4x and 8. 

The largest common divisor for 4x and 8 is 4. 

Divide 4x and 8 by 4

4x/4  =  x

8/4  =  2

Write the quotients x and 2 inside the parenthesis and multiply by the largest common divisor 4.

4(x + 2)

So, 

4x + 8  =  4(x + 2)

2. Answer : 

16a + 64b - 4c

Find the largest common divisors for 16a, 64b and 4c. 

The largest common divisor for  16a, 64b and 4c is 4. 

Divide 16a, 64b and 4c by 4 

16a/4  =  4a, 64b / 4  =  16b and 4c / 4  =  c

Write the quotients 4a, 16b and c inside the parenthesis and multiply by the largest common divisor 4.

4(4a + 16b - c)

So,

16a + 64b - 4c  =  4(4a + 16b - c) 

3. Answer : 

-25x + 15

Find the largest common divisors for 25x and 15. 

The largest common divisor for 25x and 15 is 5. 

Divide 25x and 15 by 5.

25x/5  =  5x

15/5   =  3

Write the quotients 5x and 3 inside the parenthesis and multiply by the largest common divisor 5x.

5(-5x + 3)

So,

-25x + 15  =  5(-5x + 3)

4. Answer : 

1.5y - 9

Find the largest common divisors for 1.5y and  9.

The largest common divisor for 1.5y and 9. is 1.5

Divide each term by 1.5y.

1.5y/1.5  =  y 

9/1.5  =  6

Write the quotients y and 6 inside the parenthesis and multiply by the largest common divisor 1.5.

1.5(y - 6)

So, 

1.5y - 9  =  1.5(y - 6)

5. Answer : 

4w - 8x + 16y - 32z

Find the largest common divisors for 4w, 8x, 16y and 32z.

The largest common divisor 4w, 8x, 16y and 32z is 4.

Divide each term by 4.

4w/4  =  w

8x/4  =  2x

16y/4  =  4y

32z/4  =  8z

Write the quotients inside the parenthesis and multiply by the largest common divisor 4.

4(w - 2x + 4y - 8z)

So, 

4w - 8x + 16y - 32z  =  4(w - 2x + 4y - 8z)

6. Answer :

8x + 3y

It cannot be factorized, because we don't see the common factor for 8 and 3.

7. Answer :

5k² - 40k + 10

Each coefficient is a multiple of 5, factoring 5 we get

= 5(k² - 8k + 2)

We cannot factorize here after.

8. Answer :

-x² + 16

Factoring negative, we get

= -(x² - 16)

= -(x² - 4²)

Looks like a² - b² = (a + b)(a - b)

x² - 4² = (x + 4)(x - 4)

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

9. Answer :

5p⁶ + 20p⁴ - 50p³

Every coefficient is a multiple of 5, factoring 5 we get

= 5p³ (p³ + 4p - 10)

10. Answer :

Area of the rectangle = 12x² - 21x - 6

Width = 3x - 6

To find the length of the rectangle, we may have to factorize this polynomial.

12x² - 21x - 6 = 3(4x² - 7x - 2)

= 3(4x² - 8x + 1x - 2)

= 3[4x(x - 2) + 1(x - 2)]

= 3[(4x + 1)(x - 2)]

= (4x + 1) [3 (x - 2)]

= (4x + 1) (3x - 6)

Here 3x - 6 is the width of the rectangle, then length of the rectangle is 4x + 1.

11. Answer :

Volume of rectangular prism = 6c³ - 22c² - 8c

height = 2c

Factoring 2c, we get

= 2c (3c² - 11c - 4)

= 2c (3c² - 12c + 1c - 4)

= 2c [3c(c - 4) + 1(c - 4)]

= 2c (3c + 1)(c - 4)

Length of = 3c + 1 and width = c - 4

12. Answer :

Area = y² - 14y + 45

Width = y - 9

Area of rectangle = length x width

y² - 14y + 45 = length x (y - 9)

By factoring this quadratic function, we get

y² - 9y - 5y + 45 = length x (y - 9)

y(y - 9) - 5(y - 9) = length x (y - 9)

(y - 9)(y - 5) = length x (y - 9)

Dividing by y - 9, we get

y - 5 = width

So, width of the rectangle is y - 5.

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