Maths-
General
Easy

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)
    = a2 - ab + ab - b2
    = a2 - b2
    Final Answer:
    The product of (𝑎 + 𝑏) and (𝑎 − 𝑏) is a2 - b2 which has only two terms and hence it is a binomial.

    Related Questions to study

    General
    Maths-

    Graph the equation straight Y equals negative 0.5 straight x

    Step by step solution:
    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 straight Y equals negative 0.5 straight x

    Maths-General
    Step by step solution:
    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.
     

    General
    Maths-

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

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

    • Step by step explanation: 
      • Given:
    (x + 3) (x + 4)
    • Step 1:
    Product
    (x + 3) (x + 4)
    x (x + 4) + 3 (x + 4)
    [ a ( b + c) = ab + ac ]
    x2 + 4x + 3x + 12
    x2 + 7x + 12

    Hence, constant term is 12
    • Final Answer: 
    Correct option B. 12.

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

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

    • Step by step explanation: 
      • Given:
    (x + 3) (x + 4)
    • Step 1:
    Product
    (x + 3) (x + 4)
    x (x + 4) + 3 (x + 4)
    [ a ( b + c) = ab + ac ]
    x2 + 4x + 3x + 12
    x2 + 7x + 12

    Hence, constant term is 12
    • Final Answer: 
    Correct option B. 12.
    General
    Maths-

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

    (3𝑦 − 5)(3𝑦 + 5)= 3𝑦(3𝑦 + 5) − 5(3𝑦 + 5)

    =  3𝑦(3y) + 3𝑦(5) - 5(3𝑦) - 5(5)

    = 9𝑦2 + 15𝑦 - 15𝑦 - 25

    = 9𝑦2 - 25
    Final Answer:
    Hence, the simplified form of (3𝑦 − 5)(3𝑦 + 5) is 9𝑦2 - 25.
     

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

    Maths-General
    (3𝑦 − 5)(3𝑦 + 5)= 3𝑦(3𝑦 + 5) − 5(3𝑦 + 5)

    =  3𝑦(3y) + 3𝑦(5) - 5(3𝑦) - 5(5)

    = 9𝑦2 + 15𝑦 - 15𝑦 - 25

    = 9𝑦2 - 25
    Final Answer:
    Hence, the simplified form of (3𝑦 − 5)(3𝑦 + 5) is 9𝑦2 - 25.
     

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    General
    Maths-

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

    (𝑥 − 4)(𝑥 + 4) = 𝑥(𝑥 + 4) − 4(𝑥 + 4)
    =  𝑥(𝑥) + 𝑥(4) - 4(𝑥) - 4(4)
    = 𝑥2 + 4𝑥 - 4𝑥 - 16
    = 𝑥2 - 16
    Final Answer:
    Hence, the simplified form of (𝑥 − 4)(𝑥 + 4) is 𝑥2 - 16
     

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

    Maths-General
    (𝑥 − 4)(𝑥 + 4) = 𝑥(𝑥 + 4) − 4(𝑥 + 4)
    =  𝑥(𝑥) + 𝑥(4) - 4(𝑥) - 4(4)
    = 𝑥2 + 4𝑥 - 4𝑥 - 16
    = 𝑥2 - 16
    Final Answer:
    Hence, the simplified form of (𝑥 − 4)(𝑥 + 4) is 𝑥2 - 16
     
    General
    Maths-

    The area of the rectangle is 𝑥2 + 11𝑥 + 28. Its length is x + __ and its width is __+ 4. Find the missing terms in the length and the width.

    Answer:
    • 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:
    Area of triangle = x2 + 11x + 28
    • Step 1:
    • Factorise x2 + 11x + 28.
    So,
    x2 + 11x + 28
    split the middle term
    x2 + 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.
    Length is given by (x + _)
    And width is given by (_ + 4)
    Hence,
    Length is x + 7
    Width is x + 4.
    • Final Answer:
    Hence, missing terms in length and width are 7 and x respectively.

    The area of the rectangle is 𝑥2 + 11𝑥 + 28. Its length is x + __ and its width is __+ 4. Find the missing terms in the length and the width.

    Maths-General
    Answer:
    • 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:
    Area of triangle = x2 + 11x + 28
    • Step 1:
    • Factorise x2 + 11x + 28.
    So,
    x2 + 11x + 28
    split the middle term
    x2 + 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.
    Length is given by (x + _)
    And width is given by (_ + 4)
    Hence,
    Length is x + 7
    Width is x + 4.
    • Final Answer:
    Hence, missing terms in length and width are 7 and x respectively.
    General
    Maths-

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

    Answer:
    • Hint:
    ○    Perform basic arithmetic operations to simplify expression.
    • Step by step explanation:
    ○    Given:
    Expression:
    12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]
    ○    Step 1:
    ○    Simplify:
    12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]
    rightwards double arrow12- [13a - 20a + 28 - 8 {2a - 17a}]
    rightwards double arrow12- [- 7a + 28 - 8 {- 15a}]
    rightwards double arrow12- [- 7a + 28 - {- 120a}]
    rightwards double arrow12- [- 7a + 28 + 120a}]
    rightwards double arrow12- 28 + 113a
    rightwards double arrow113a - 16
    • Final Answer:
    113a - 16

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

    Maths-General
    Answer:
    • Hint:
    ○    Perform basic arithmetic operations to simplify expression.
    • Step by step explanation:
    ○    Given:
    Expression:
    12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]
    ○    Step 1:
    ○    Simplify:
    12 - [13a - 4(5a -7) - 8 {2a -(20a - 3a)}]
    rightwards double arrow12- [13a - 20a + 28 - 8 {2a - 17a}]
    rightwards double arrow12- [- 7a + 28 - 8 {- 15a}]
    rightwards double arrow12- [- 7a + 28 - {- 120a}]
    rightwards double arrow12- [- 7a + 28 + 120a}]
    rightwards double arrow12- 28 + 113a
    rightwards double arrow113a - 16
    • Final Answer:
    113a - 16
    parallel
    General
    Maths-

    Write the product in standard form. (2𝑥 + 5)2

    (2𝑥 + 5)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)

    = 4x2 + 10x + 10x + 25

    = 4x2 + 20x + 25
    Final Answer:
    Hence, the simplified value of (2𝑥 + 5)2 is  4x2 + 20x + 25.
     

    Write the product in standard form. (2𝑥 + 5)2

    Maths-General
    (2𝑥 + 5)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)

    = 4x2 + 10x + 10x + 25

    = 4x2 + 20x + 25
    Final Answer:
    Hence, the simplified value of (2𝑥 + 5)2 is  4x2 + 20x + 25.
     

    General
    Maths-

    Write the product in standard form. (𝑥 − 7)2

    (𝑥 − 7)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)

    = x2 - 7x - 7x + 72

    = x2 - 14x + 49
    Final Answer:
    Hence, the simplified value of (𝑥 − 7)2 is   x2 - 14x + 49.
     

    Write the product in standard form. (𝑥 − 7)2

    Maths-General
    (𝑥 − 7)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)

    = x2 - 7x - 7x + 72

    = x2 - 14x + 49
    Final Answer:
    Hence, the simplified value of (𝑥 − 7)2 is   x2 - 14x + 49.
     

    General
    Maths-

    (𝑥 + 9)(𝑥 + 9) =

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

    =  x(x) +  x(9) + 9(x) + 9(9)

    = x2 + 9x + 9x + 92

    = x2 + 18x + 81
    Final Answer:
    Hence, the simplified value of (𝑥 + 9)(𝑥 + 9) is  x2 + 18x + 81.
     

    (𝑥 + 9)(𝑥 + 9) =

    Maths-General
    (𝑥 + 9)(𝑥 + 9) = x(x + 9) + 9(x + 9)

    =  x(x) +  x(9) + 9(x) + 9(9)

    = x2 + 9x + 9x + 92

    = x2 + 18x + 81
    Final Answer:
    Hence, the simplified value of (𝑥 + 9)(𝑥 + 9) is  x2 + 18x + 81.
     

    parallel
    General
    Maths-

    Find the area of the rectangle.

    Answer:
    • 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 + 4) × (4x - 2)
              Area of rectangle = 2x × (4x - 2) + 4 × (4x - 2)
              [ a (b + c) = ab + ac ]
              Area of rectangle = 8x2 - 4x + 16x - 8
              Area of rectangle = 8x2 + 12x - 8
              • Final Answer:
              Hence, area of rectangle is 8x2 + 12x - 8.

    Find the area of the rectangle.

    Maths-General
    Answer:
    • 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 + 4) × (4x - 2)
              Area of rectangle = 2x × (4x - 2) + 4 × (4x - 2)
              [ a (b + c) = ab + ac ]
              Area of rectangle = 8x2 - 4x + 16x - 8
              Area of rectangle = 8x2 + 12x - 8
              • Final Answer:
              Hence, area of rectangle is 8x2 + 12x - 8.

    General
    Maths-

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

    Hint:
    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 open parentheses x subscript 1 comma y subscript 1 close parentheses and open parentheses x subscript 2 comma y subscript 2 close parentheses satisfying the equation, the slope is given by
    m equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction
    Taking any two points from the table and denoting them as
    open parentheses x subscript 1 comma y subscript 1 close parentheses equals left parenthesis 2 comma 25 right parenthesis text  and  end text open parentheses x subscript 2 comma y subscript 2 close parentheses equals left parenthesis 4 comma 50 right parenthesis, we have
    m equals fraction numerator 50 minus 25 over denominator 4 minus 2 end fraction
    m equals 25 over 2 equals 12.5
    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.

    Maths-General
    Hint:
    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 open parentheses x subscript 1 comma y subscript 1 close parentheses and open parentheses x subscript 2 comma y subscript 2 close parentheses satisfying the equation, the slope is given by
    m equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction
    Taking any two points from the table and denoting them as
    open parentheses x subscript 1 comma y subscript 1 close parentheses equals left parenthesis 2 comma 25 right parenthesis text  and  end text open parentheses x subscript 2 comma y subscript 2 close parentheses equals left parenthesis 4 comma 50 right parenthesis, we have
    m equals fraction numerator 50 minus 25 over denominator 4 minus 2 end fraction
    m equals 25 over 2 equals 12.5
    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.
    General
    Maths-

    Simplify: 4x2(7x -5) -6x2(2 -4x)+ 18x3

    Answer:
    • Hint:
    ○    Group the like term.
    ○    Like terms are those whose coefficients are the same.
    ○    The term is decreasing in order of their power.
    • Step by step explanation:
    ○    Given:
    Expression: 4x2(7x -5) -6x2(2 -4x)+ 18x3
    ○    Step 1:
    ○    simplify:
    4x2(7x -5) -6x2(2 -4x)+ 18x3
    rightwards double arrow 28x3 - 20x2 - 12x2 + 24x3 + 18x3
    ○    Step 2:
    ○     Arrange the term in decreasing order of their power.
    rightwards double arrow 28x3 + 24x3 + 18x3 - 20x2 - 12x2
    rightwards double arrow70x3 - 32x2
    • Final Answer:
    70x3 - 32x2

    Simplify: 4x2(7x -5) -6x2(2 -4x)+ 18x3

    Maths-General
    Answer:
    • Hint:
    ○    Group the like term.
    ○    Like terms are those whose coefficients are the same.
    ○    The term is decreasing in order of their power.
    • Step by step explanation:
    ○    Given:
    Expression: 4x2(7x -5) -6x2(2 -4x)+ 18x3
    ○    Step 1:
    ○    simplify:
    4x2(7x -5) -6x2(2 -4x)+ 18x3
    rightwards double arrow 28x3 - 20x2 - 12x2 + 24x3 + 18x3
    ○    Step 2:
    ○     Arrange the term in decreasing order of their power.
    rightwards double arrow 28x3 + 24x3 + 18x3 - 20x2 - 12x2
    rightwards double arrow70x3 - 32x2
    • Final Answer:
    70x3 - 32x2
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    General
    Maths-

    (𝑎 + (−3))2 =

    (𝑎 + (−3))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)

    = a2 - 3a - 3a + 32

    = a2 - 6a + 9
    Final Answer:
    The simplified value of (𝑎 + (−3))2 is  a2 - 6a + 9. Hence, option b is correct.
     

    (𝑎 + (−3))2 =

    Maths-General
    (𝑎 + (−3))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)

    = a2 - 3a - 3a + 32

    = a2 - 6a + 9
    Final Answer:
    The simplified value of (𝑎 + (−3))2 is  a2 - 6a + 9. Hence, option b is correct.
     

    General
    Maths-

    Use the table method to multiply a binomial with a trinomial.
    (−3𝑥2 + 1) (2𝑥2 + 3𝑥 − 4)

    Answer:
      • Given:
    (-3x2 + 1) (2x2 + 3x - 4)
    • Step 1:

    • Step 2:
    Add all terms:
    (-6x4 – 9x3 + 12x2 + 2x2 + 3x – 4)
    = -6x4 – 9x3 + 14x2 +3x-4
    Hence,
    (-3x2 + 1) (2x2 + 3x – 4) = -6x4 – 9x3 + 14x2 +3x-4
    • Final Answer: 
    -6x4 – 9x3 + 14x2 +3x-4

    Use the table method to multiply a binomial with a trinomial.
    (−3𝑥2 + 1) (2𝑥2 + 3𝑥 − 4)

    Maths-General
    Answer:
      • Given:
    (-3x2 + 1) (2x2 + 3x - 4)
    • Step 1:

    • Step 2:
    Add all terms:
    (-6x4 – 9x3 + 12x2 + 2x2 + 3x – 4)
    = -6x4 – 9x3 + 14x2 +3x-4
    Hence,
    (-3x2 + 1) (2x2 + 3x – 4) = -6x4 – 9x3 + 14x2 +3x-4
    • Final Answer: 
    -6x4 – 9x3 + 14x2 +3x-4
    General
    Maths-

    (𝑥 − 2)2 =

    (𝑥 − 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)

    = 𝑥2 - 2𝑥 - 2𝑥 + 22

    = 𝑥2 - 4𝑥 + 4
    Final Answer:
    The simplified value of (𝑥 − 2)2 is 𝑥2 - 4𝑥 + 4. Hence, option d is correct.
     

    (𝑥 − 2)2 =

    Maths-General
    (𝑥 − 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)

    = 𝑥2 - 2𝑥 - 2𝑥 + 22

    = 𝑥2 - 4𝑥 + 4
    Final Answer:
    The simplified value of (𝑥 − 2)2 is 𝑥2 - 4𝑥 + 4. Hence, option d is correct.
     

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