Question

# Why the product of two binomials (𝑎 + 𝑏) and (𝑎 − 𝑏) is a binomial instead of a trinomial?

Hint:

### A polynomial equation with two terms usually joined by a plus or minus sign is called a binomial.

The methods used to find the product of binomials are called special products.

Difference of squares is a case of a special product which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign.

## The correct answer is: a2 - b2

### Let’s first the product of (𝑎 + 𝑏) and (𝑎 − 𝑏)

(a + b)(a - b) = a(a - b) + b(a - b)

= a(a) + a(-b) + b(a) + b(-b)

= a^{2} - ab + ab - b^{2}

= a^{2} - b^{2}

Final Answer:

The product of (𝑎 + 𝑏) and (𝑎 − 𝑏) is a^{2} - b^{2} which has only two terms and hence it is a binomial.

### Related Questions to study

### Graph the equation

The given equation is

Y = -0.5x

First we make a table of points satisfying the equation.

Putting x = 0 in the above equation, we will get, y = 0

Similarly, putting x = 1, in the above equation, we get y = -0.5

Continuing this way, we have

For x = -1, we get y = 0.5

For x = 2, we get y = -1

For x = -2, we get y = 1

Making a table of all these points, we have

Now, we plot these points on the graph

After plotting the points, we join them with a line to get the graph of the equation.

### Graph the equation

The given equation is

Y = -0.5x

First we make a table of points satisfying the equation.

Putting x = 0 in the above equation, we will get, y = 0

Similarly, putting x = 1, in the above equation, we get y = -0.5

Continuing this way, we have

For x = -1, we get y = 0.5

For x = 2, we get y = -1

For x = -2, we get y = 1

Making a table of all these points, we have

Now, we plot these points on the graph

After plotting the points, we join them with a line to get the graph of the equation.

### The constant term in the product (𝑥 + 3) (𝑥 + 4) is

- Hints:
- Multiplication polynomials
- Multiply each term with each other.
- Distributive identity:
- a × (b + c) = ab + ac

- Step by step explanation:
- Given:

- Step 1:

(x + 3) (x + 4)

x (x + 4) + 3 (x + 4)

[ a ( b + c) = ab + ac ]

x

^{2}+ 4x + 3x + 12

x

^{2}+ 7x + 12

Hence, constant term is 12

- Final Answer:

### The constant term in the product (𝑥 + 3) (𝑥 + 4) is

- Hints:
- Multiplication polynomials
- Multiply each term with each other.
- Distributive identity:
- a × (b + c) = ab + ac

- Step by step explanation:
- Given:

- Step 1:

(x + 3) (x + 4)

x (x + 4) + 3 (x + 4)

[ a ( b + c) = ab + ac ]

x

^{2}+ 4x + 3x + 12

x

^{2}+ 7x + 12

Hence, constant term is 12

- Final Answer:

### Write the product in standard form. (3𝑦 − 5)(3𝑦 + 5)

= 3𝑦(3y) + 3𝑦(5) - 5(3𝑦) - 5(5)

= 9𝑦^{2} + 15𝑦 - 15𝑦 - 25

= 9𝑦^{2} - 25

Final Answer:

Hence, the simplified form of (3𝑦 − 5)(3𝑦 + 5) is 9𝑦^{2} - 25.

### Write the product in standard form. (3𝑦 − 5)(3𝑦 + 5)

= 3𝑦(3y) + 3𝑦(5) - 5(3𝑦) - 5(5)

= 9𝑦^{2} + 15𝑦 - 15𝑦 - 25

= 9𝑦^{2} - 25

Final Answer:

Hence, the simplified form of (3𝑦 − 5)(3𝑦 + 5) is 9𝑦^{2} - 25.

### Write the product in standard form. (𝑥 − 4)(𝑥 + 4)

= 𝑥(𝑥) + 𝑥(4) - 4(𝑥) - 4(4)

= 𝑥

^{2}+ 4𝑥 - 4𝑥 - 16

= 𝑥

^{2}- 16

Final Answer:

Hence, the simplified form of (𝑥 − 4)(𝑥 + 4) is 𝑥

^{2}- 16

### Write the product in standard form. (𝑥 − 4)(𝑥 + 4)

= 𝑥(𝑥) + 𝑥(4) - 4(𝑥) - 4(4)

= 𝑥

^{2}+ 4𝑥 - 4𝑥 - 16

= 𝑥

^{2}- 16

Final Answer:

Hence, the simplified form of (𝑥 − 4)(𝑥 + 4) is 𝑥

^{2}- 16

### The area of the rectangle is 𝑥^{2} + 11𝑥 + 28. Its length is x + __ and its width is __+ 4. Find the missing terms in the length and the width.

- Hint:
- Area of rectangle is given by product of length and breadth.
- Area of rectangle = length × breadth.
- Area of rectangle = l × b
- Step by step explanation:

- Given:

^{2}+ 11x + 28

- Step 1:
- Factorise x
^{2}+ 11x + 28.

x

^{2}+ 11x + 28

split the middle term

x

^{2}+ 7x + 4x + 28

take common terms

x(x+7) + 4(x + 7)

(x+ 4) (x + 7)

Hence, the above two terms be length and width of rectangle.

- Step 2:
- Compare terms with length and breadth.

And width is given by (_ + 4)

Hence,

Length is x + 7

Width is x + 4.

- Final Answer:

### The area of the rectangle is 𝑥^{2} + 11𝑥 + 28. Its length is x + __ and its width is __+ 4. Find the missing terms in the length and the width.

- Hint:
- Area of rectangle is given by product of length and breadth.
- Area of rectangle = length × breadth.
- Area of rectangle = l × b
- Step by step explanation:

- Given:

^{2}+ 11x + 28

- Step 1:
- Factorise x
^{2}+ 11x + 28.

x

^{2}+ 11x + 28

split the middle term

x

^{2}+ 7x + 4x + 28

take common terms

x(x+7) + 4(x + 7)

(x+ 4) (x + 7)

Hence, the above two terms be length and width of rectangle.

- Step 2:
- Compare terms with length and breadth.

And width is given by (_ + 4)

Hence,

Length is x + 7

Width is x + 4.

- Final Answer:

### Simplify: 12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]

- Hint:

- Step by step explanation:

Expression:

12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]

○ Step 1:

○ Simplify:

12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]

12- [13a - 20a + 28 - 8 {2a - 17a}]

12- [- 7a + 28 - 8 {- 15a}]

12- [- 7a + 28 - {- 120a}]

12- [- 7a + 28 + 120a}]

12- 28 + 113a

113a - 16

- Final Answer:

### Simplify: 12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]

- Hint:

- Step by step explanation:

Expression:

12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]

○ Step 1:

○ Simplify:

12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]

12- [13a - 20a + 28 - 8 {2a - 17a}]

12- [- 7a + 28 - 8 {- 15a}]

12- [- 7a + 28 - {- 120a}]

12- [- 7a + 28 + 120a}]

12- 28 + 113a

113a - 16

- Final Answer:

### Write the product in standard form. (2𝑥 + 5)^{2}

^{2}can be written as (2𝑥 + 5)(2𝑥 + 5)

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

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

= 4x^{2} + 10x + 10x + 25

= 4x^{2} + 20x + 25

Final Answer:

Hence, the simplified value of (2𝑥 + 5)^{2} is 4x^{2} + 20x + 25.

### Write the product in standard form. (2𝑥 + 5)^{2}

^{2}can be written as (2𝑥 + 5)(2𝑥 + 5)

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

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

= 4x^{2} + 10x + 10x + 25

= 4x^{2} + 20x + 25

Final Answer:

Hence, the simplified value of (2𝑥 + 5)^{2} is 4x^{2} + 20x + 25.

### Write the product in standard form. (𝑥 − 7)^{2}

^{2}can be written as (𝑥 − 7)(𝑥 − 7)

(𝑥 − 7)(𝑥 − 7) = x(x - 7) - 7(x - 7)

= x(x) + x(-7) - 7(x) - 7(-7)

= x^{2} - 7x - 7x + 7^{2}

= x^{2} - 14x + 49

Final Answer:

Hence, the simplified value of (𝑥 − 7)^{2} is x^{2} - 14x + 49.

### Write the product in standard form. (𝑥 − 7)^{2}

^{2}can be written as (𝑥 − 7)(𝑥 − 7)

(𝑥 − 7)(𝑥 − 7) = x(x - 7) - 7(x - 7)

= x(x) + x(-7) - 7(x) - 7(-7)

= x^{2} - 7x - 7x + 7^{2}

= x^{2} - 14x + 49

Final Answer:

Hence, the simplified value of (𝑥 − 7)^{2} is x^{2} - 14x + 49.

### (𝑥 + 9)(𝑥 + 9) =

= x(x) + x(9) + 9(x) + 9(9)

= x^{2} + 9x + 9x + 9^{2}

= x^{2} + 18x + 81

Final Answer:

Hence, the simplified value of (𝑥 + 9)(𝑥 + 9) is x^{2} + 18x + 81.

### (𝑥 + 9)(𝑥 + 9) =

= x(x) + x(9) + 9(x) + 9(9)

= x^{2} + 9x + 9x + 9^{2}

= x^{2} + 18x + 81

Final Answer:

Hence, the simplified value of (𝑥 + 9)(𝑥 + 9) is x^{2} + 18x + 81.

### Find the area of the rectangle.

- Hint:

- Area of rectangle is given by product of length and breadth.
- Area of rectangle = length × breadth.
- Area of rectangle = l × b
- Distributive property: a (b + c) = ab + ac
- Step by step explanation:
- Given:
- Length (l) = 2x + 4 unitsBreadth (b) = 4x - 2 units
- Step 1:
- Find area of rectangle.So,

Area of rectangle = l × b

Area of rectangle = (2x + 4) × (4x - 2)- Step 1:
- Product of polynomials.

Area of rectangle = 2x × (4x - 2) + 4 × (4x - 2)

[ a (b + c) = ab + ac ]

Area of rectangle = 8x^{2}- 4x + 16x - 8

Area of rectangle = 8x^{2}+ 12x - 8- Final Answer:

^{2}+ 12x - 8.

### Find the area of the rectangle.

- Hint:

- Area of rectangle is given by product of length and breadth.
- Area of rectangle = length × breadth.
- Area of rectangle = l × b
- Distributive property: a (b + c) = ab + ac
- Step by step explanation:
- Given:
- Length (l) = 2x + 4 unitsBreadth (b) = 4x - 2 units
- Step 1:
- Find area of rectangle.So,

Area of rectangle = l × b

Area of rectangle = (2x + 4) × (4x - 2)- Step 1:
- Product of polynomials.

Area of rectangle = 2x × (4x - 2) + 4 × (4x - 2)

[ a (b + c) = ab + ac ]

Area of rectangle = 8x^{2}- 4x + 16x - 8

Area of rectangle = 8x^{2}+ 12x - 8- Final Answer:

^{2}+ 12x - 8.

### The table below shows the distance a train traveled over time. How can you determine the equation that represents this relationships.

We are given the relationship between time and distance travelled by a train. We find the equation representing this relationship in the slope-intercept form which is y = mx + c. First we find the slope (m) of the equation from the given data and then find the y-intercept (b). We then use these two values to get the final equation.

Step by step solution:

We denote the time on the x-axis and distance on the y-axis.

Let the equation be denoted by y = mx + c

First, we find the slope of the equation.

For two points and satisfying the equation, the slope is given by

Taking any two points from the table and denoting them as

, we have

Next, for finding the y-intercept, we input any value from the table in the equation

y = mx + c

We choose the first point (2,25), and using it in the above equation, we have

25 = m.2 + c

Putting the value of m in the above relation, we have

25 = 12.5 × 2 + c

Simplifying, we have

25 = 25 + c

Thus, we get

c = 25 - 25 = 0

So, the y-intercept c = 0

Using the value of m and c, we get the required equation,

y = 12.5x + 0

That is

y = 12.5x

*Note:*

Instead of using these particular points to find the slope and the y-intercept, we can use any other points from the table; we will get the same equation at the end. We can also verify this equation by inserting the points from the table and checking if the equation is satisfied. Finally, we can find the line in any other forms of a straight line.

### The table below shows the distance a train traveled over time. How can you determine the equation that represents this relationships.

We are given the relationship between time and distance travelled by a train. We find the equation representing this relationship in the slope-intercept form which is y = mx + c. First we find the slope (m) of the equation from the given data and then find the y-intercept (b). We then use these two values to get the final equation.

Step by step solution:

We denote the time on the x-axis and distance on the y-axis.

Let the equation be denoted by y = mx + c

First, we find the slope of the equation.

For two points and satisfying the equation, the slope is given by

Taking any two points from the table and denoting them as

, we have

Next, for finding the y-intercept, we input any value from the table in the equation

y = mx + c

We choose the first point (2,25), and using it in the above equation, we have

25 = m.2 + c

Putting the value of m in the above relation, we have

25 = 12.5 × 2 + c

Simplifying, we have

25 = 25 + c

Thus, we get

c = 25 - 25 = 0

So, the y-intercept c = 0

Using the value of m and c, we get the required equation,

y = 12.5x + 0

That is

y = 12.5x

*Note:*

Instead of using these particular points to find the slope and the y-intercept, we can use any other points from the table; we will get the same equation at the end. We can also verify this equation by inserting the points from the table and checking if the equation is satisfied. Finally, we can find the line in any other forms of a straight line.

### Simplify: 4x^{2}(7x -5) -6x^{2}(2 -4x)+ 18x^{3}

- Hint:

○ Like terms are those whose coefficients are the same.

○ The term is decreasing in order of their power.

- Step by step explanation:

Expression: 4x

^{2}(7x -5) -6x

^{2}(2 -4x)+ 18x

^{3}

○ Step 1:

○ simplify:

4x

^{2}(7x -5) -6x

^{2}(2 -4x)+ 18x

^{3}

28x

^{3}- 20x

^{2}- 12x

^{2}+ 24x

^{3}+ 18x

^{3}

○ Step 2:

○ Arrange the term in decreasing order of their power.

28x

^{3 }+ 24x

^{3}+ 18x

^{3}- 20x

^{2}- 12x

^{2}

70x

^{3}- 32x

^{2}

- Final Answer:

^{3}- 32x

^{2}

### Simplify: 4x^{2}(7x -5) -6x^{2}(2 -4x)+ 18x^{3}

- Hint:

○ Like terms are those whose coefficients are the same.

○ The term is decreasing in order of their power.

- Step by step explanation:

Expression: 4x

^{2}(7x -5) -6x

^{2}(2 -4x)+ 18x

^{3}

○ Step 1:

○ simplify:

4x

^{2}(7x -5) -6x

^{2}(2 -4x)+ 18x

^{3}

28x

^{3}- 20x

^{2}- 12x

^{2}+ 24x

^{3}+ 18x

^{3}

○ Step 2:

○ Arrange the term in decreasing order of their power.

28x

^{3 }+ 24x

^{3}+ 18x

^{3}- 20x

^{2}- 12x

^{2}

70x

^{3}- 32x

^{2}

- Final Answer:

^{3}- 32x

^{2}

### (𝑎 + (−3))^{2} =

^{2}can be written as (𝑎 + (−3))(𝑎 + (−3))

(𝑎 + (−3))(𝑎 + (−3)) = a(𝑎 + (−3)) − 3(𝑎 + (−3))

= a(a) + a(-3) - 3(a) - 3(-3)

= a^{2} - 3a - 3a + 3^{2}

= a^{2} - 6a + 9

Final Answer:

The simplified value of (𝑎 + (−3))^{2} is a^{2} - 6a + 9. Hence, option b is correct.

### (𝑎 + (−3))^{2} =

^{2}can be written as (𝑎 + (−3))(𝑎 + (−3))

(𝑎 + (−3))(𝑎 + (−3)) = a(𝑎 + (−3)) − 3(𝑎 + (−3))

= a(a) + a(-3) - 3(a) - 3(-3)

= a^{2} - 3a - 3a + 3^{2}

= a^{2} - 6a + 9

Final Answer:

The simplified value of (𝑎 + (−3))^{2} is a^{2} - 6a + 9. Hence, option b is correct.

### Use the table method to multiply a binomial with a trinomial.

(−3𝑥^{2} + 1) (2𝑥^{2} + 3𝑥 − 4)

- Given:

^{2}+ 1) (2x

^{2}+ 3x - 4)

- Step 1:

- Step 2:

(-6x

^{4}– 9x

^{3}+ 12x

^{2}+ 2x

^{2}+ 3x – 4)

= -6x

^{4}– 9x

^{3}+ 14x

^{2 }+3x-4

Hence,

(-3x

^{2}+ 1) (2x

^{2}+ 3x – 4) = -6x

^{4}– 9x

^{3}+ 14x

^{2 }+3x-4

- Final Answer:

^{4}– 9x

^{3}+ 14x

^{2 }+3x-4

### Use the table method to multiply a binomial with a trinomial.

(−3𝑥^{2} + 1) (2𝑥^{2} + 3𝑥 − 4)

- Given:

^{2}+ 1) (2x

^{2}+ 3x - 4)

- Step 1:

- Step 2:

(-6x

^{4}– 9x

^{3}+ 12x

^{2}+ 2x

^{2}+ 3x – 4)

= -6x

^{4}– 9x

^{3}+ 14x

^{2 }+3x-4

Hence,

(-3x

^{2}+ 1) (2x

^{2}+ 3x – 4) = -6x

^{4}– 9x

^{3}+ 14x

^{2 }+3x-4

- Final Answer:

^{4}– 9x

^{3}+ 14x

^{2 }+3x-4

### (𝑥 − 2)^{2} =

^{2}can be written as (𝑥 − 2)(𝑥 − 2)

(𝑥 − 2)(𝑥 − 2) = 𝑥(𝑥 − 2) − 2(𝑥 − 2)

= 𝑥(𝑥) + 𝑥(-2) - 2(𝑥) - 2(-2)

= 𝑥^{2} - 2𝑥 - 2𝑥 + 2^{2}

= 𝑥^{2} - 4𝑥 + 4

Final Answer:

The simplified value of (𝑥 − 2)^{2} is 𝑥^{2} - 4𝑥 + 4. Hence, option d is correct.

### (𝑥 − 2)^{2} =

^{2}can be written as (𝑥 − 2)(𝑥 − 2)

(𝑥 − 2)(𝑥 − 2) = 𝑥(𝑥 − 2) − 2(𝑥 − 2)

= 𝑥(𝑥) + 𝑥(-2) - 2(𝑥) - 2(-2)

= 𝑥^{2} - 2𝑥 - 2𝑥 + 2^{2}

= 𝑥^{2} - 4𝑥 + 4

Final Answer:

The simplified value of (𝑥 − 2)^{2} is 𝑥^{2} - 4𝑥 + 4. Hence, option d is correct.