FACTORING POLYNOMIALS WITH 4 TERMS BY GROUPING WORKSHEET

Factor the following polynomials by grouping :

Question 1 :

x³ - 2x² - x + 2

Solution :

= x³ - 2x² - x + 2

= (x³ - 2x²) + (-x + 2)

 = x²(x - 2) - 1(x - 2)

 =  (x² - 1)(x - 2)

 =  (x² - 1²)(x - 2)

Using algebraic identity a² - b² = (a + b)(a -b),

= (x + 1)(x - 1)(x - 2)

Question 2 :

x³ + 3x² - x - 3

Solution :

= x³ + 3x² - x - 3

= (x³ + 3x²) + (-x - 3)

= x²(x + 3) - 1(x + 3)

= (x² - 1)(x + 3)

= (x² - 1²)(x + 3)

Using algebraic identity a² - b² = (a + b)(a -b),

= (x + 1)(x - 1)(x + 3)

Question 3 :

x³ + x² - 4x - 4

Solution :

= x³ + x² - 4x - 4

= (x³ + x²) + (-4x - 4)

= x²(x + 1) - 4(x - 1)

= (x² - 4)(x + 1)

= (x² - 2²)(x + 1)

Using algebraic identity a² - b² = (a + b)(a -b),

= (x + 2)(x - 2)(x + 1)

Question 4 :

x³ - 3x² + 2x - 6

Solution :

= x³ - 3x² + 2x - 6

= (x³ - 3x²) + (2x - 6)

= x²(x - 3) + 2(x - 3)

= (x² + 2)(x - 3)

Question 5 :

x⁴ - x³ - x + x²

Solution :

= x⁴ - x³ - x + x²

Arrange the terms with powers in descending order. 

= x⁴ - x³ + x² - x

= (x⁴ - x³) + (x² - x)

= x³(x - 1) + x(x - 1)

= (x³ + x)(x - 1)

= x(x² + 1)(x - 1)

Question 6 :

x⁸ - x⁴ - x² + 1

Solution :

= x⁸ - x⁴ + x² + 1

= (x⁸ - x⁴) + (x² + 1)

= x⁴(x⁴ - 1) + 1(x² + 1)

= x⁴(x⁴ - 1) + 1(x² + 1)

= x⁴[(x²)² - 1²] + 1(x² + 1)

= x⁴(x² - 1)(x² + 1) + 1(x² + 1)

= (x² + 1)[x⁴(x² - 1) + 1]

Question 7 :

5a - 5b - xa + xb

Solution :

= 5a - 5b - xa + xb

= (5a - 5b) + (-xa + xb)

= 5(a - b) - x(a - b)

= (a - b)(5 - x)

Question 8 :

2xy - 4x - 3ay + 6a

Solution :

= 2xy - 4x - 3ay + 6a

= (2xy - 4x) + (-3ay + 6a)

= 2x(y - 2) - 3a(y - 2)

= (2x - 3a)(y - 2)

Question 9 :

y³ - y² + 2y - 2

Solution :

= y³ - y² + 2y - 2

Factoring y² from the first two terms and factoring 2 from the last two terms.

= y²(y - 1) + 2(y - 1)

= (y² + 2) (y - 1)

So, the factors are (y² + 2) (y - 1).

Question 10 :

p² q - 25q  + 3p² - 75

Solution :

= p² q - 25q  + 3p² - 75

= q (p² - 25)  + 3(p² - 25)

= (q + 3) (p² - 25)

= (q + 3) (p² - 5²)

= (q + 3)(p + 5)(p - 5)

So, the factors are (q + 3)(p + 5)(p - 5).

Question 11 :

2xy - x²y - 6 + 3x

Solution :

= 2xy - x²y - 6 + 3x

= xy(2 - x) - 3(2 - x)

= (xy - 3) (2 - x)

Question 12 :

x³ + 3x² + 2x + 6

Solution :

= x³ + 3x² + 2x + 6

= x² (x + 3) + 2(x + 3)

= (x² + 2)(x + 3)

Question 13 :

ay - yx - x² + ax

Solution :

= ay - yx - x² + ax

= y(a - x) - x(x - a)

Factoring negative from x - a, we get

= y(a - x) + x(a - x)

= (y + x)(a - x)

Question 14 :

a³ + 3a² + a + 3

Solution :

= a³ + 3a² + a + 3

= a² (a + 3) + 1(a + 3)

= (a² + 1)(a + 3)

Question 15 :

y² + 2y + yx + 2x

Solution :

= y² + 2y + yx + 2x

= y(y + 2) + x(y + 2)

= (y + 2)(y + x)

Question 16 :

The volume (in cubic feet) of a room in the shape of a rectangular prism is represented by 12z³ − 27z. Find expressions that could represent the dimensions of the room.

Solution :

= 12z³ − 27z

= 3z(4z² - 9)

= 3z(4z² - 3²)

= 3z ((2z)² - 3²)

= 3z (2z + 3)(2z - 3)

Question 17 :

How can you use the factored form of the polynomial

x⁴ − 2x³ − 9x² + 18x = x(x − 3)(x + 3)(x − 2)

to find the x-intercepts of the graph of the function?

Solution :

x⁴ − 2x³ − 9x² + 18x = x(x − 3)(x + 3)(x − 2)

Let f(x) = y = x⁴ − 2x³ − 9x² + 18x

To find x-intercept, we will apply y = 0

x(x − 3)(x + 3)(x − 2) = 0

Equating each linear factor to 0, we get

x = 0, x - 3 = 0, x + 3 = 0 and x - 2 = 0

x = 0, x = 3, x = -3 and x = 2.

So, the x-intercepts are -3, 0, 2 and 3.

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