Extracting the common factor in parentheses - Examples, Exercises and Solutions

Common Factor Extraction Method:
Identify the largest free number that we can extract.
Then, let's move on to the variables and ask what is the least number of times the X X appears in any element?
Multiply the free number by the variable the same number of times we have found and we will obtain the greatest common factor.

To verify that you have correctly extracted the common factor, open the parentheses and see if you have returned to the original exercise.

Suggested Topics to Practice in Advance

  1. Factoring using contracted multiplication

Practice Extracting the common factor in parentheses

Examples with solutions for Extracting the common factor in parentheses

Exercise #1

Solve the following problem:

x2x=0 x^2-x=0

Video Solution

Step-by-Step Solution

Shown below is the given equation:

x2x=0 x^2-x=0

First note that on the left side we are able to factor the expression using a common factor. The largest common factor for the numbers and letters in this case is x x and this is due to the fact that the first power is the lowest power in the equation. Therefore it is included both in the term with the second power and in the term with the first power. Any power higher than this is not included in the term with the first power, which is the lowest. Hence this is the term with the highest power that can be factored out as a common factor from all terms in the expression. Continue to factor the expression:

x2x=0x(x1)=0 x^2-x=0 \\ \downarrow\\ x(x-1)=0

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore given that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

x=0 \boxed{x=0}

or:

x1=0x=1 x-1=0\\ \downarrow\\ \boxed{x=1}

Let's summarize then the solution to the equation:

x2x=0x(x1)=0x=0x=0x1=0x=1x=0,1 x^2-x=0 \\ \downarrow\\ x(x-1)=0 \\ \downarrow\\ x=0 \rightarrow\boxed{ x=0}\\ x-1=0 \rightarrow \boxed{x=1}\\ \downarrow\\ \boxed{x=0,1}

Therefore the correct answer is answer B.

Answer

x=0,1 x=0,1

Exercise #2

x4+2x2=0 x^4+2x^2=0

Video Solution

Step-by-Step Solution

To solve the equation x4+2x2=0x^4 + 2x^2 = 0, we will use the technique of factoring. Let's proceed step-by-step:

First, notice that both terms x4x^4 and 2x22x^2 have a common factor of x2x^2. We can factor x2x^2 out from the equation:

x2(x2+2)=0x^2(x^2 + 2) = 0

Now, to solve for xx, we apply the Zero Product Property, which gives us that at least one of the factors must be zero:

  • x2=0x^2 = 0 or
  • x2+2=0x^2 + 2 = 0

Solving the first case, x2=0x^2 = 0:

x=0x = 0

For the second case, x2+2=0x^2 + 2 = 0, we reach:

x2=2x^2 = -2

Since x2=2x^2 = -2 has no real solutions (squares of real numbers are non-negative), we can conclude that this equation doesn't provide additional real solutions.

Therefore, the only real solution to the given equation is x=0x = 0.

The correct choice from the provided options is:

x=0 x=0

Answer

x=0 x=0

Exercise #3

Solve the following problem:

x7x6=0 x^7-x^6=0

Video Solution

Step-by-Step Solution

Shown below is the given problem:

x7x6=0 x^7-x^6=0

First, note that on the left side we are able factor the expression by using a common factor.

The largest common factor for the numbers and letters in this case is x6 x^6 due to the fact that the sixth power is the lowest power in the equation . Therefore it is included both in the term with the seventh power and in the term with the sixth power. Any power higher than this is not included in the term with the sixth power, which is the lowest. Hence this is the term with the highest power that can be factored out as a common factor from all terms in the expression. Continue to factor the expression.

x7x6=0x6(x1)=0 x^7-x^6=0 \\ \downarrow\\ x^6(x-1)=0

Proceed to the left side of the equation that we obtained from the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore due to the fact that the only way to obtain 0 from multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

x6=0/6x=±0x=0 x^6=0 \hspace{8pt}\text{/}\sqrt[6]{\hspace{6pt}}\\ x=\pm0\\ \boxed{x=0}

(in this case taking the even root of the right side of the equation will yield two possibilities - positive and negative however given that we're dealing with zero, we only obtain one answer)

or:

x1=0x=1 x-1=0\\ \downarrow\\ \boxed{x=1}

Let's summarize the solution of the equation:

x7x6=0x6(x1)=0x6=0x=0x1=0x=1x=0,1 x^7-x^6=0 \\ \downarrow\\ x^6(x-1)=0 \\ \downarrow\\ x^6=0 \rightarrow\boxed{ x=0}\\ x-1=0 \rightarrow \boxed{x=1}\\ \downarrow\\ \boxed{x=0,1}

Therefore the correct answer is answer C.

Answer

x=0,1 x=0,1

Exercise #4

4x412x3=0 4x^4-12x^3=0

Solve the equation above for x.

Video Solution

Step-by-Step Solution

Shown below is the given problem:

4x412x3=0 4x^4-12x^3=0

First, note that on the left side we are able factor the expression by using a common factor. The largest common factor for the numbers and variables in this case is 4x3 4x^3 due to the fact that the third power is the lowest power in the equation. Therefore it is included in both the term with the fourth power and the term with the third power. Any power higher than this is not included in the term with the third power, which is the lowest. Hence this is the term with the highest power that can be factored out as a common factor for the variables,

For the numbers, note that 12 is a multiple of 4, therefore 4 is the largest common factor for the numbers in both terms of the expression,

Let's continue to factor the expression:

4x412x3=04x3(x3)=0 4x^4-12x^3=0 \\ \downarrow\\ 4x^3(x-3)=0

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore given that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

4x3=0/:4x3=0/3x=0 4x^3=0 \hspace{8pt}\text{/}:4\\ x^3=0 \hspace{8pt}\text{/}\sqrt[3]{\hspace{6pt}}\\ \boxed{x=0}

In solving the equation above, we first divided both sides of the equation by the term with the unknown and then extracted a cube root for both sides of the equation.

(In this case, extracting an odd root for the right side of the equation yielded one possibility)

Or:

x3=0x=3 x-3=0\\ \boxed{x=3}

Let's summarize the solution of the equation:

4x412x3=04x3(x3)=04x3=0x=0x3=0x=3x=0,3 4x^4-12x^3=0 \\ \downarrow\\ 4x^3(x-3)=0 \\ \downarrow\\ 4x^3=0 \rightarrow\boxed{ x=0}\\ x-3=0\rightarrow \boxed{x=3}\\ \downarrow\\ \boxed{x=0,3}

Therefore the correct answer is answer A.

Answer

x=0,3 x=0,3

Exercise #5

Solve the following problem:

3x2+9x=0 3x^2+9x=0

Video Solution

Step-by-Step Solution

Shown below is the given problem:

3x2+9x=0 3x^2+9x=0

First, note that in the left side we are able to factor the expression by using a common factor. The largest common factor for the numbers and letters in this case is 3x 3x due to the fact that the first power is the lowest power in the equation and therefore is included both in the term with the second power and in the term with the first power. Any power higher than this is not included in the term with the first power, which is the lowest. Therefore this is the term with the highest power that can be factored out as a common factor from all terms for the letters,

For the numbers, note that 9 is a multiple of 3, therefore it is the largest common factor for the numbers in both terms of the expression,

Let's continue to factor the expression:

3x2+9x=03x(x+3)=0 3x^2+9x=0 \\ \downarrow\\ 3x(x+3)=0

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore, given that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

3x=0/:3x=0 3x=0 \hspace{8pt}\text{/}:3\\ \boxed{x=0}

In solving the above equation, we divided both sides of the equation by the term with the variable,

Or:

x+3=0x=3 x+3=0 \\ \boxed{x=-3}

Let's summarize the solution of the equation:

3x2+9x=03x(x+3)=03x=0x=0x+3=0x=3x=0,3 3x^2+9x=0 \\ \downarrow\\ 3x(x+3)=0 \\ \downarrow\\ 3x=0 \rightarrow\boxed{ x=0}\\ x+3=0\rightarrow \boxed{x=-3}\\ \downarrow\\ \boxed{x=0,-3}

Therefore the correct answer is answer C.

Answer

x=0,x=3 x=0,x=-3

Exercise #6

Solve the following problem:

x54x4=0 x^5-4x^4=0

Video Solution

Step-by-Step Solution

Shown below is the given problem:

x54x4=0 x^5-4x^4=0

Note that on the left side we are able to factor the expression by using a common factor.

The largest common factor for the numbers and variables in this case is x4 x^4 since the fourth power is the lowest power in the equation and therefore is included both in the term with the fifth power and in the term with the fourth power. Any power higher than this is not included in the term with the fourth power, which is the lowest, and therefore this is the term with the highest power that can be factored out as a common factor from all terms in the expression. Proceed to factor the expression.

x54x4=0x4(x4)=0 x^5-4x^4=0 \\ \downarrow\\ x^4(x-4)=0

Let's continue to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore, due to the fact that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

meaning:

x4=0/4x=±0x=0 x^4=0 \hspace{8pt}\text{/}\sqrt[4]{\hspace{6pt}}\\ x=\pm0\\ \boxed{x=0} (In this case taking the even root of the right side of the equation will yield two possibilities - positive and negative, however since we're dealing with zero, we only obtain one solution)

Or:

x4=0x=4 x-4=0\\ \downarrow\\ \boxed{x=4}

Let's summarize the solution of the equation:

x54x4=0x4(x4)=0x4=0x=0x4=0x=4x=0,4 x^5-4x^4=0 \\ \downarrow\\ x^4(x-4)=0 \\ \downarrow\\ x^4=0 \rightarrow\boxed{ x=0}\\ x-4=0 \rightarrow \boxed{x=4}\\ \downarrow\\ \boxed{x=0,4}

Therefore the correct answer is answer C.

Answer

x=4,x=0 x=4,x=0

Exercise #7

x64x4=0 x^6-4x^4=0

Video Solution

Step-by-Step Solution

To solve this problem, we start by factoring the given equation:

The equation is x64x4=0 x^6 - 4x^4 = 0 . Notice that both terms contain a power of x x . We can factor out the greatest common factor, which is x4 x^4 .

This yields x4(x24)=0 x^4(x^2 - 4) = 0 .

Next, we apply the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero:

  • First factor: x4=0 x^4 = 0 . This implies that x=0 x = 0 .
  • Second factor: x24=0 x^2 - 4 = 0 . We solve this quadratic equation by factoring further:

The quadratic equation x24=0 x^2 - 4 = 0 can be factored using the difference of squares:

(x2)(x+2)=0 (x - 2)(x + 2) = 0 .

Again applying the zero-product property, we set each factor equal to zero:

  • For x2=0 x - 2 = 0 , x=2 x = 2 .
  • For x+2=0 x + 2 = 0 , x=2 x = -2 .

Thus, the complete set of solutions to the equation is x=0,x=2,x=2 x = 0, x = 2, x = -2 .

Therefore, the solution to the problem is x=0,x=±2 x = 0, x = \pm 2 .

Answer

x=0,x=±2 x=0,x=\pm2

Exercise #8

Solve the following problem:

x6+x5=0 x^6+x^5=0

Video Solution

Step-by-Step Solution

Shown below is the given equation:

x6+x5=0 x^6+x^5=0

First, note that on the left side we are able to factor the expression by using a common factor.

The largest common factor for the numbers and variables in this case is x5 x^5 given that the fifth power is the lowest power in the equation and therefore is included both in the term with the sixth power and in the term with the fifth power. Any power higher than this is not included in the term with the fifth power, which is the lowest, and therefore this is the term with the highest power that can be factored out as a common factor from all terms in the expression. Proceed with the factoring of the expression:

x6+x5=0x5(x+1)=0 x^6+x^5=0 \\ \downarrow\\ x^5(x+1)=0

Let's continue to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore, due to the fact that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

meaning:

x5=0/5x=0 x^5=0 \hspace{8pt}\text{/}\sqrt[5]{\hspace{6pt}}\\ \boxed{x=0} (in this case taking the odd root of the right side of the equation will yield one possibility)

or:

x+1=0x=1 x+1=0\\ \boxed{x=-1}

Let's summarize the solution of the equation:

x6+x5=0x5(x+1)=0x5=0x=0x+1=0x=1x=0,1 x^6+x^5=0 \\ \downarrow\\ x^5(x+1)=0 \\ \downarrow\\ x^5=0 \rightarrow\boxed{ x=0}\\ x+1=0 \rightarrow \boxed{x=-1}\\ \downarrow\\ \boxed{x=0,-1}

Therefore the correct answer is answer A.

Answer

x=1,x=0 x=-1,x=0

Exercise #9

x4+x2=0 x^4+x^2=0

Video Solution

Step-by-Step Solution

The problem at hand is to solve the equation x4+x2=0 x^4 + x^2 = 0 .

Let's begin by factoring the expression:

The given equation is: x4+x2=0 x^4 + x^2 = 0

We can factor out the common factor of x2 x^2 from both terms:

x2(x2+1)=0 x^2(x^2 + 1) = 0

To solve for x x , we set each factor equal to zero:

  • x2=0 x^2 = 0

Solving for x x , we have:

x=0 x = 0

Next, consider the second factor:

  • x2+1=0 x^2 + 1 = 0

Solving for x x , we have:

x2=1 x^2 = -1

Since x2=1 x^2 = -1 has no real solutions, we ignore these solutions in the real number system.

Thus, the only real solution to the equation x4+x2=0 x^4 + x^2 = 0 is:

x=0 x = 0

Answer

x=0 x=0

Exercise #10

Solve the following equation:

7x1014x9=0 7x^{10}-14x^9=0

Video Solution

Step-by-Step Solution

Shown below is the given equation:

7x1014x9=0 7x^{10}-14x^9=0

First, note that on the left side we are able to factor the expression using a common factor.

The largest common factor for the numbers and variables in this case is 7x9 7x^9 given that the ninth power is the lowest power in the equation and therefore is included in both the term with the tenth power and the term with the ninth power. Any power higher than this is not included in the term with the ninth power, which is the lowest, and therefore this is the term with the highest power that can be factored out as a common factor from all terms for the variables,

For the numbers, note that 14 is a multiple of 7, therefore 7 is the largest common factor for the numbers in both terms of the expression,

Let's continue and perform the factoring:

7x1014x9=07x9(x2)=0 7x^{10}-14x^9=0 \\ \downarrow\\ 7x^9(x-2)=0

On the left side of the equation that we obtained in the last step there is a multiplication of algebraic expressions and on the right side the number 0, therefore, since the only way to obtain a result of 0 from a multiplication operation is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

7x9=0/:7x9=0/9x=0 7x^9=0 \hspace{8pt}\text{/}:7\\ x^9=0 \hspace{8pt}\text{/}\sqrt[9]{\hspace{6pt}}\\ \boxed{x=0}

In solving the equation above, we first divided both sides of the equation by the term with the variable, and then we proceeded to extract a ninth root from both sides of the equation.

(In this case, extracting an odd root from the right side of the equation yielded one possibility)

Or:

x2=0x=2 x-2=0 \\ \boxed{x=2}

Let's summarize the solution of the equation:

7x1014x9=07x9(x2)=07x9=0x=0x2=0x=2x=0,2 7x^{10}-14x^9=0 \\ \downarrow\\ 7x^9(x-2)=0\\ \downarrow\\ 7x^9=0 \rightarrow\boxed{ x=0}\\ x-2=0\rightarrow \boxed{x=2}\\ \downarrow\\ \boxed{x=0,2}

Therefore, the correct answer is answer A.

Answer

x=2,x=0 x=2,x=0

Exercise #11

Solve the following problem:

7x3x2=0 7x^3-x^2=0

Video Solution

Step-by-Step Solution

Solve the given equation:

7x3x2=0 7x^3-x^2=0

Note that on the left side we are able to factor the expression using a common factor. The largest common factor for the numbers and letters in this case is x2 x^2 since the second power is the lowest power in the equation and therefore is included both in the term with the third power and in the term with the second power. Any power higher than this is not included in the term with the second power, which is the lowest, and therefore this is the term with the highest power that can be factored out as a common factor from all terms in the expression.

7x3x2=0x2(7x1)=0 7x^3-x^2=0 \\ \downarrow\\ x^2(7x-1)=0

Note that the left side of the equation that we obtained in the last step is a multiplication of algebraic expressions and on the right side the number 0.

Therefore, given that the only way to obtain 0 from a multiplication operation is to multiply by 0. Hence at least one of the expressions in the multiplication on the left side must equal zero,

meaning:

x2=0/x=±0x=0 x^2=0 \hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x=\pm0\\ \boxed{x=0} (in this case taking the even root of the right side of the equation will indeed yield two possibilities, positive and negative. However since we're dealing with zero, we'll get only one possibility)

or:

7x1=0 7x-1=0 Let's solve this equation in order to obtain the additional solutions (if they exist) to the given equation:

We obtained a simple first-degree equation which we'll solve by isolating the unknown on one side, we'll do this by moving terms and then dividing both sides of the equation by the coefficient of the unknown:

7x1=07x=1/:7x=17 7x-1=0 \\ 7x=1\hspace{8pt}\text{/}:7\\ \boxed{x=\frac{1}{7}}

Let's summarize the solution of the equation:

7x3x2=0x2(7x1)=0x2=0x=07x1=0x=17x=0,17 7x^3-x^2=0 \\ \downarrow\\ x^2(7x-1)=0\\ \downarrow\\ x^2=0 \rightarrow\boxed{ x=0}\\ 7x-1=0\rightarrow \boxed{x=\frac{1}{7}}\\ \downarrow\\ \boxed{x=0,\frac{1}{7}}

Therefore the correct answer is answer C.

Answer

x=0,x=17 x=0,x=\frac{1}{7}

Exercise #12

What is the value of x?

x4x3=2x2 x^4-x^3=2x^2

Video Solution

Step-by-Step Solution

To solve the problem x4x3=2x2 x^4 - x^3 = 2x^2 , let's proceed as follows:

  • Step 1: Set the equation to zero.
    x4x32x2=0 x^4 - x^3 - 2x^2 = 0
  • Step 2: Factor out the greatest common factor.
    The common factor among all terms is x2 x^2 .
    Factoring out x2 x^2 gives:
    x2(x2x2)=0 x^2(x^2 - x - 2) = 0
  • Step 3: Solve the factors.
    This equation breaks into two factors that can be solved separately:
    • x2=0 x^2 = 0
    • x2x2=0 x^2 - x - 2 = 0
  • Step 4: Solve x2=0 x^2 = 0 .
    Since x2=0 x^2 = 0 , we get:
    x=0 x = 0
  • Step 5: Solve x2x2=0 x^2 - x - 2 = 0 .
    This can be factored further. We look for two numbers that multiply to 2-2 and add up to 1-1.
    These numbers are 2-2 and 11, so we factor as:
    (x2)(x+1)=0 (x - 2)(x + 1) = 0
  • Step 6: Solve the quadratic factors.
    Set each factor equal to zero:
    • x2=0x=2 x - 2 = 0 \Rightarrow x = 2
    • x+1=0x=1 x + 1 = 0 \Rightarrow x = -1

The solutions to the equation x4x3=2x2 x^4 - x^3 = 2x^2 are x=1,0,2 x = -1, 0, 2 .

Therefore, the correct answer is:

x=1,2,0 x = -1, 2, 0

Answer

x=1,2,0 x=-1,2,0

Exercise #13

Solve the following problem:

15x430x3=0 15x^4-30x^3=0

Video Solution

Step-by-Step Solution

Shown below is the given equation:

15x430x3=0 15x^4-30x^3=0

First, note that on the left side we are able to factor the expression by using a common factor. The largest common factor for the numbers and variables in this case is 15x3 15x^3 given that the third power is the lowest power in the equation. Therefore it is included in both the term with the fourth power and the term with the third power. Any power higher than this is not included in the term with the third power, which is the lowest. Hence this is the term with the highest power that can be factored out as a common factor from all terms for the variables,

For the numbers, note that 30 is a multiple of 15, therefore 15 is the largest common factor for the numbers for both terms in the expression,

Let's continue to factor the expression:

15x430x3=015x3(x2)=0 15x^4-30x^3=0 \\ \downarrow\\ 15x^3(x-2)=0

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore due to the fact that the only way to obtain 0 as a result of multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

15x3=0/:15x3=0/3x=0 15x^3=0 \hspace{8pt}\text{/}:15\\ x^3=0 \hspace{8pt}\text{/}\sqrt[3]{\hspace{6pt}}\\ \boxed{x=0}

In solving the equation above, we first divided both sides of the equation by the term with the unknown and then proceeded to extract a cube root for both sides of the equation.

(In this case extracting an odd root for the right side of the equation yielded one possibility)

Or:

x2=0x=2 x-2=0\\ \boxed{x=2}

Let's summarize the solution of the equation:

15x430x3=015x3(x2)=015x3=0x=0x2=0x=2x=0,2 15x^4-30x^3=0 \\ \downarrow\\ 15x^3(x-2)=0 \\ \downarrow\\ 15x^3=0 \rightarrow\boxed{ x=0}\\ x-2=0\rightarrow \boxed{x=2}\\ \downarrow\\ \boxed{x=0,2}

Therefore the correct answer is answer B.

Answer

x=0,2 x=0,2

Exercise #14

Solve for x:

7x514x4=0 7x^5-14x^4=0

Video Solution

Step-by-Step Solution

Shown below is the given equation:

7x514x4=0 7x^5-14x^4=0

First, note that on the left side we are able to factor the expression using a common factor. The largest common factor for the numbers and variables in this case is 7x4 7x^4 due to the fact that the fourth power is the lowest power in the equation. Therefore it is included both in the term with the fifth power and the term with the fourth power. Any power higher than this is not included in the term with the fourth power, which is the lowest. Hence this is the term with the highest power that can be factored out as a common factor from all terms for the variables,

For the numbers, note that 14 is a multiple of 7, therefore 7 is the largest common factor for the numbers in both terms of the expression,

Let's continue to factor the expression:

7x514x4=07x4(x2)=0 7x^5-14x^4=0 \\ \downarrow\\ 7x^4(x-2)=0

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore due to the fact that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

7x4=0/:7x4=0/4x=±0x=0 7x^4=0 \hspace{8pt}\text{/}:7\\ x^4=0 \hspace{8pt}\text{/}\sqrt[4]{\hspace{6pt}}\\ x=\pm0\\ \boxed{x=0}

In solving the equation above, we first divided both sides of the equation by the term with the variable, and then we took the fourth root of both sides of the equation.

(In this case, taking an even root of the right side of the equation will yield two possibilities - positive and negative however given that we're dealing with zero, we only obtain one answer)

Or:

x2=0x=2 x-2=0\\ \boxed{x=2}

Let's summarize the solution of the equation:

7x514x4=07x4(x2)=07x4=0x=0x2=0x=2x=0,2 7x^5-14x^4=0 \\ \downarrow\\ 7x^4(x-2)=0 \\ \downarrow\\ 7x^4=0 \rightarrow\boxed{ x=0}\\ x-2=0\rightarrow \boxed{x=2}\\ \downarrow\\ \boxed{x=0,2}

Therefore the correct answer is answer A.

Answer

x=0,2 x=0,2

Exercise #15

Solve the following problem:

7x821x7=0 7x^8-21x^7=0

Video Solution

Step-by-Step Solution

Shown below is the given equation:

7x821x7=0 7x^8-21x^7=0

First, note that on the left side we are able factor the expression by using a common factor. The largest common factor for the numbers and variables in this case is 7x7 7x^7 due to the fact that the seventh power is the lowest power in the equation. Therefore it is included in both the term with the eighth power and the term with the seventh power. Any power higher than this is not included in the term with the seventh power, which is the lowest. Hence this is the term with the highest power that can be factored out as a common factor for variables,

For the numbers, we notice that 21 is a multiple of 7, therefore 7 is the largest common factor for numbers in both terms of the expression,

Let's continue to factor the expression:

7x821x7=07x7(x3)=0 7x^8-21x^7=0 \\ \downarrow\\ 7x^7(x-3)=0

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore, given that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

7x7=0/:7x7=0/7x=0 7x^7=0 \hspace{8pt}\text{/}:7\\ x^7=0 \hspace{8pt}\text{/}\sqrt[7]{\hspace{6pt}}\\ \boxed{x=0}

In solving the equation above, we first divided both sides of the equation by the term with the variable and then extracted a seventh root from both sides of the equation.

(In this case, extracting an odd root from the right side of the equation yielded one possibility)

Or:

x3=0x=3 x-3=0\\ \boxed{x=3}

Let's summarize the solution of the equation:

7x821x7=07x7(x3)=07x7=0x=0x3=0x=3x=0,3 7x^8-21x^7=0 \\ \downarrow\\ 7x^7(x-3)=0 \\ \downarrow\\ 7x^7=0 \rightarrow\boxed{ x=0}\\ x-3=0\rightarrow \boxed{x=3}\\ \downarrow\\ \boxed{x=0,3}

Therefore, the correct answer is answer B.

Answer

x=0,3 x=0,3