Finding the Unknown: Remainder and Factor Theorem Applications

Remainder and Factor Theorem Applications

Number Theory

The product of three binomials is 2x³+ 7x² – 7x – 12. If one binomial is x + 1, what are the other two binomials?

Solution:

x = -1

Apply the Synthetic Division:

	2	7	-7	-12
-1	2	5	-12	0

= 2x²+ 5x - 12

Use the Quadratic Formula:

2x² + 5x – 12

a =2, b = 5, c = -12

x = -b ± √b² - 4ac / 2a

x = -5 ± √5² – 4(2)(-12) / 2(2)

x = -5 ± √25 + 96 / 4

x = -5 ± √121 / 4

x₁ = -5 + 11 / 4

x₁= 6/4

x₁= 3/2

x₂= -5 – 11 / 4

x₂= -16/4

x₂ = -4

Answer: The other two binomials are (2x - 3) and (x + 4).

Check:

(x + 1) (2x - 3) (x + 4) ≟ 2x³+ 7x² – 7x – 12

(2x²– x - 3)(x + 4) ≟ 2x³+ 7x² – 7x – 12

2x³+ 7x² – 7x – 12 = 2x³+ 7x² – 7x – 12

Therefore, 2x³+ 7x² – 7x – 12 is the product of the three binomials (x + 1), (2x - 3), and

(x + 4).

Number Theory

Martha represented three consecutive even integers in a unique way. If the product of the three integers is represented by x³ – 12x² + 44x – 48, and one integer is represented by
x – 4, what expressions did Martha use for the other integers? Could she have used a different set of expressions?

Solution:

x = 4

Apply the Synthetic Division:

	1	-12	44	-48
4	1	-8	12	0

= x² – 8x + 12

Factor:
x² – 8x + 12

= (x – 6)(x – 2)

x = 6

x = 2

Answer: Martha used the expression (x - 6) and (x - 2) for the other integers. Yes, she have used a different set of expressions.

Check:

(x – 4)(x – 6)(x – 2) ≟ x³ – 12x² + 44x – 48

(x²– 10x + 24)(x - 2) ≟ x³ – 12x² + 44x – 48

x³ – 12x² + 44x – 48 = x³ – 12x² + 44x – 48

Therefore, x³ – 12x² + 44x – 48 is the product of the three consecutive even integers (x – 4), (x – 6), and (x – 2) Martha represented.

Number Theory

Hector represented the product of four numbers as x⁴ – x³ – 10x² – 8x. One number was represented by x and another by x + 1. What expressions were used to represent the other two numbers?

Solution:

Divide first by x:

x⁴ – x³ – 10x² – 8x / x

= x³– x² – 10x – 8

x = -1

Apply the Synthetic Division:

	1	-1	-10	-8
-1	1	-2	-8	0

= x² – 2x – 8

Factor:
x² – 2x – 8

= (x + 2)(x – 4)

x = -2

x = 4

The expressions (x + 2) and (x - 4) were used to represent the other two numbers.

Check:
x(x + 1)(x + 2)(x – 4) ≟ x⁴ – x³ – 10x² – 8x

(x²+ x)(x + 2)(x – 4) ≟ x⁴ – x³ – 10x² – 8x

(x³+ 3x² + 2x)(x – 4) ≟ x⁴ – x³ – 10x² – 8x

x⁴ – x³ – 10x² – 8x = x⁴ – x³ – 10x² – 8x

Therefore, x⁴ – x³ – 10x² – 8x is the product of four numbers x, (x + 1), (x + 2), and

(x – 4) represented by Hector.

Geometry

If the volume of a box is represented by the expression (x³ – 3x²– 10x + 24) cm³ and its width by

(x - 2) cm, what binomials can be used to represent the other two dimensions?

Solution:

x = 2

Apply the Synthetic Division:

	1	-3	-10	24
2	1	-1	-12	0

= x²– x – 12

Factor:
x²– x – 12

= (x + 3)(x -4)

x = -3

x = 4

Answer: The binomials (x + 3) cm and (x -4) cm can be used to represent the other two dimensions.

Check:

(x - 2)(x + 3)(x - 4) ≟ x³ – 3x²– 10x + 24

(x² + x – 6)(x – 4) ≟ x³ – 3x²– 10x + 24

x³ – 3x²– 10x + 24 = x³ – 3x²– 10x + 24

Therefore, x³ – 3x²– 10x + 24 cm³ is the expression that represents the volume of the box with dimensions of (x - 2) cm, (x + 3) cm,and (x - 4) cm.

Geometry

If the volume of a box is represented by the expression (a³x³ – 6a²x² + 11ax – 6) cm³and its length is

(ax – 1) cm, what binomials can be used to represent the other two dimensions?

Solution:

Divide:

a³x³ – 6a²x² + 11ax – 6 / ax - 1
= a²x²- 5ax + 6

Factor:

a²x²- 5ax + 6

= (ax-3)(ax-2)

ax = 3

ax = 2

Answer: The binomials (ax - 3) cm and (ax - 2) cm can be used to represent the other two dimensions.

Check:

(ax - 1)(ax - 3)(ax - 2) ≟ a³x³ – 6a²x² + 11ax – 6

(a²x²- 4ax + 3)(ax - 2) ≟ a³x³ – 6a²x² + 11ax – 6

a³x³ – 6a²x² + 11ax – 6 = a³x³ – 6a²x² + 11ax – 6

Therefore, (a³x³ – 6a²x² + 11ax – 6) cm³ is the expression that represents the volume of a box with dimensions of (ax - 1) cm, (ax - 3) cm, and (ax - 2) cm.

More Applications

1. The product of three binomials is
3x³ + x²– 27x - 9. If one binomial is x - 3, what are the other two binomials?

x = 3

Apply the Synthetic Division:

	3	1	-27	-9
3	3	10	3	0

= 3x² + 10x + 3

Factor:

3x² + 10x + 3

= (3x + 1)(x + 3)

x = -1/3

x = -3

Answer: The other two binomials are (3x + 1) and (x + 3).

Check:

(x - 3)(x + 3)(3x + 1) ≟ 3x³ + x²– 27x - 9

(x² - 9)(3x + 1) ≟ 3x³ + x²– 27x - 9

3x³ + x²– 27x - 9 = 3x³ + x²– 27x - 9

Therefore, (x - 3), (x + 3), and (3x + 1) are factors of 3x³ + x²– 27x - 9.

2. The product of three binomials is 2x³ - x²– 8x + 4. If one binomial is x - 2, what are the other two binomials?

x = 2

Apply the Synthetic Division:

	2	-1	-8	4
2	2	3	-2	0

= 2x²+ 3x -2

Factor:

2x²+ 3x -2

= (2x – 1)(x + 2)

x = 1/2

x = -2

Answer: The other two binomials are (2x - 1) and (x + 2).

Check:
(x - 2)(2x - 1)(x + 2) ≟ 2x³ - x²– 8x + 4
(2x²- 5x + 2)(x + 2) ≟ 2x³ - x²– 8x + 4
2x³ - x²– 8x + 4 = 2x³ - x²– 8x + 4

Therefore, (x - 2), (2x - 1), and (x + 2) are factors of 2x³ - x²– 8x + 4.

3. The product of three binomials is y³ - 4y²– 7y + 10. If one binomial is y – 1, what are the other two binomials?

y = 1

Apply the Synthetic Division:

	1	-4	-7	10
1	1	-3	-10	0

= y²– 3y -10

Factor:

y²– 3y -10

= (y + 2)(y - 5)

y = -2

y = 5

Answer: The other two binomials are (y + 2) and (y - 5).

Check:
(y - 1)(y + 2)(y - 5) ≟ y³ - 4y²– 7y + 10
(y²+ y - 2)(y - 5) ≟ y³ - 4y²– 7y + 10
y³ - 4y²– 7y + 10 = y³ - 4y²– 7y + 10

Therefore, y³ - 4y²– 7y + 10 is the product of the binomials (y - 1), (y + 2), and (y - 5).

4. Veah represented the product of four numbers as y⁴ – 3y³ – 10y² + 24y. One number was represented by y and another by y - 4. What expressions were used to represent the other two numbers?

Divide first by y:

y⁴ – 3y³ – 10y² + 24y / y

= y³ – 3y² – 10y + 24

y = 4

Apply the Synthetic Division:

	1	-3	-10	24
4	1	1	-6	0

= y² + y – 6

Factor:

y² + y – 6

= (y +3)(y-2)

y = -3

y = 2

Answer: The expressions (y + 3) and (y - 2) were used to represent the other two numbers.

Check:
y(y - 4)(y +3)(y-2) ≟ y⁴ – 3y³ – 10y² + 24y
(y²- 4y)(y +3)(y-2) ≟ y⁴ – 3y³ – 10y² + 24y
(y³ – y² - 12y)(y - 2) ≟ y⁴ – 3y³ – 10y² + 24y
y⁴ – 3y³ – 10y² + 24y = y⁴ – 3y³ – 10y² + 24y

Therefore, y⁴ – 3y³ – 10y² + 24y is the product of the four integers y, (y - 4), (y + 3), and (y - 2) Veah represented.

5. Three numbers were represented by (x - 4), (x + 3), and (x -2). How should the product be represented?

Multiply:
(x - 4)(x + 3)(x - 2) = (x²– x - 12)(x - 2)
(x - 4)(x + 3)(x - 2) = x³- 3x²– 10x + 24

Answer: The product should be represented as x³- 3x²– 10x + 24.

Check:
x³- 3x²– 10x + 24 / (x - 4)(x + 3)(x - 2)

Use the Synthetic Division:

	1	-3	-10	24
4	1	1	-6	0

= x²+ x - 6

Factor:
x²+ x - 6
= (x + 3)(x - 2)

x = -3

x = 2

Therefore, (x - 4), (x + 3), and (x -2) are factors of x³- 3x²– 10x + 24.

“Prime numbers are what is left when you have taken all the patterns away. I think prime numbers are like life. They are very logical, but you could never work out the rules, even if you spent all your time thinking about them.”
― Mark Haddon

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Friday, October 4, 2013

Remainder and Factor Theorem Applications

Remainder and Factor Theorem Applications

Number Theory

Number Theory

Number Theory

Geometry

Geometry

More Applications