Examples with solutions for Factorization - Common Factor: Matching expressions equal in value

Exercise #1

Which of the expressions is equivalent to the expression?

8x2+4x 8x^2+4x

Video Solution

Step-by-Step Solution

Let's solve the problem step by step:

  • Step 1: Identify the greatest common factor (GCF) of the coefficients in the expression 8x2+4x8x^2 + 4x. The numbers are 8 and 4, and the GCF is 4.

  • Step 2: Factor out the common variable. Both terms have xx as a common variable factor, so the GCF of the variable part is xx.

  • Step 3: Factor the expression using the GCF. We take 4x4x as a common factor from both terms:
        8x28x^2 can be rewritten as 4x2x4x \cdot 2x.
        +4x+4x can be rewritten as 4x14x \cdot 1.

  • Step 4: Write the factored expression:
        8x2+4x=4x(2x+1)8x^2 + 4x = 4x(2x + 1).

  • Step 5: Verify by checking each option. The expression we obtained 4x(2x+1)4x(2x + 1) matches the choice with 2.

Therefore, the equivalent expression is 4x(2x+1)4x(2x+1).

Answer

4x(2x+1) 4x(2x+1)

Exercise #2

Which of the expressions is equivalent to the expression?

2a(b+3)+4(b+3) 2a(b+3)+4(b+3)

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the common factor in the expression.
  • Step 2: Factor out the common factor using the distributive property.
  • Step 3: Simplify the expression inside the parentheses.

Now, let's work through each step:

Step 1: Identify the common factor. The given expression is 2a(b+3)+4(b+3) 2a(b+3) + 4(b+3) . Notice that both terms have a common factor, which is (b+3) (b+3) .

Step 2: Factor out the common factor. Using the distributive property in reverse, we can factor out (b+3) (b+3) :

2a(b+3)+4(b+3)=(b+3)(2a+4) 2a(b+3) + 4(b+3) = (b+3)(2a + 4)

Step 3: Simplify the expression inside the parentheses if needed. In this case, 2a+4 2a + 4 is already simplified.

Therefore, the expression 2a(b+3)+4(b+3) 2a(b+3) + 4(b+3) simplifies to the equivalent expression (b+3)(2a+4) (b+3)(2a+4) .

The correct choice that corresponds to this expression is choice 3: (b+3)(2a+4) (b+3)(2a+4) .

Answer

(b+3)(2a+4) (b+3)(2a+4)

Exercise #3

Which of the expressions is equivalent to the expression?

2xy+x2+3x 2xy+x^2+3x

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the greatest common factor (GCF) in the expression.

  • Step 2: Factor out the GCF from the expression.

  • Step 3: Compare the factored expression with the choices provided.

Now, let's work through each step:
Step 1: The expression given is 2xy+x2+3x 2xy + x^2 + 3x . The GCF of these terms is x x because it appears in each term.
Step 2: Factor out x x from each term, which gives:x(2y+x+3) x(2y + x + 3) This rewrites the expression in its factored form.
Step 3: Compare the factored form x(2y+x+3) x(2y + x + 3) with the answer choices.

Choice 1: 2x+2y+3 2x + 2y + 3 does not match the factored form.
Choice 2: x(2y+x+3) x(2y + x + 3) exactly matches the factored form.
Choice 3: 2x(y+4) 2x(y + 4) does not match the factored form.
Choice 4: 2x(y+x+3) 2x(y + x + 3) does not match the factored form.

Therefore, the expression 2xy+x2+3x 2xy + x^2 + 3x is equivalent to x(2y+x+3) x(2y + x + 3) .

Answer

x(2y+x+3) x(2y+x+3)

Exercise #4

Which of the expressions is equivalent to the expression?

7z+10b+2bz+35 7z+10b+2bz+35

Video Solution

Step-by-Step Solution

To solve this problem, we'll focus on factorization by grouping:

  • Step 1: Identify groupable terms in 7z+10b+2bz+35 7z + 10b + 2bz + 35 .
  • Step 2: Reorganize and group terms for greatest common factor extraction.
  • Step 3: Factor each group and simplify.

Now, let's work through each step:

Step 1:
Observe that we can reorganize the expression to facilitate grouping:

7z+35+10b+2bz 7z + 35 + 10b + 2bz .

Step 2:
Group into pairs: (7z+35)+(10b+2bz)(7z + 35) + (10b + 2bz).
Within each pair, extract common factors:

=7(z+5)+2b(z+5)= 7(z + 5) + 2b(z + 5), noticing that each group factors nicely.

Step 3:
Since both terms now have a common factor of (z+5)(z + 5), we can factor it out:

=(7+2b)(z+5)= (7 + 2b)(z + 5).

Therefore, the expression 7z+10b+2bz+35 7z + 10b + 2bz + 35 is equivalent to (7+2b)(z+5)(7 + 2b)(z + 5).

This matches choice 1: (7+2b)(z+5) (7+2b)(z+5) .

Answer

(7+2b)(z+5) (7+2b)(z+5)

Exercise #5

Which of the expressions is equivalent to the expression?

164c 16-4c

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the greatest common factor (GCF) of the terms in the expression 164c 16 - 4c .
  • Step 2: Factor out the GCF from both terms.
  • Step 3: Compare the factored expression with the provided choices to find the equivalent expression.

Let's work through each step:

Step 1: Identify the GCF.
The terms in the expression are 16 16 and 4c-4c. The greatest common factor of 16 16 and 4 4 (coefficient of c c ) is 4 4 .

Step 2: Factor out the GCF.
Factor 4 4 out from each term in the expression:
164c=4×44×c=4(4c) 16 - 4c = 4 \times 4 - 4 \times c = 4(4 - c) .

Step 3: Compare with the choices.
We factorized 164c 16 - 4c to get 4(4c) 4(4 - c) . Now we compare it with the provided choices:

  • Choice 1: 4(12c) 4(12 - c) does not match our expression.
  • Choice 2: 4(44c) 4(4 - 4c) is also not equivalent.
  • Choice 3: 4(2c) 4(2 - c) is not equivalent.
  • Choice 4: 4(4c) 4(4 - c) matches our expression precisely.

Therefore, the expression equivalent to 164c 16 - 4c is 4(4c) \boxed{4(4 - c)} .

Answer

4(4c) 4(4-c)

Exercise #6

Choose the expression that is equivalent to the following:

15z2+50zx 15z^2+50zx

Video Solution

Step-by-Step Solution

To solve the problem, the goal is to factor the expression 15z2+50zx 15z^2 + 50zx by finding the greatest common factor.

Step 1: Identify the greatest common factor (GCF):

  • The terms in the expression are 15z2 15z^2 and 50zx 50zx .
  • The numerical coefficients are 15 15 and 50 50 , and their GCF is 5 5 .
  • Both terms contain z z as a factor, so z z is also part of the GCF.
  • Therefore, the GCF of both terms is 5z 5z .

Step 2: Factor the expression using the GCF:

  • Divide each term by the GCF 5z 5z :
  • 15z25z=3z \frac{15z^2}{5z} = 3z
  • 50zx5z=10x \frac{50zx}{5z} = 10x
  • Thus, the expression can be rewritten as the product of the GCF and the remaining terms: 5z(3z+10x) 5z(3z + 10x) .

Therefore, the expression 15z2+50zx 15z^2 + 50zx is equivalent to 5z(3z+10x) 5z(3z + 10x) .

Answer

5z(3z+10x) 5z(3z+10x)

Exercise #7

Which expression is equivalent in value to the following:

99ab2+81b 99ab^2+81b

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the greatest common factor (GCF) of the coefficients in the expression.

  • Step 2: Factor out the GCF from the entire expression.

  • Step 3: Confirm that the factored expression matches one of the given answer choices.

Now, let's work through each step:
Step 1: The coefficients of the terms 99ab2 99ab^2 and 81b 81b are 99 and 81, respectively. The greatest common factor of these coefficients is 9.
Step 2: Both terms, 99ab2 99ab^2 and 81b 81b , contain the variable b b . We can factor out b b as well. Thus, the GCF of both terms in the expression 99ab2+81b 99ab^2 + 81b is 9b 9b .
Step 3: Factor 9b 9b out of both terms:
99ab2+81bamp;=9b(99ab29b+81b9b)amp;=9b(11ab+9) \begin{aligned} 99ab^2 + 81b &= 9b( \frac{99ab^2}{9b} + \frac{81b}{9b} )\\ &= 9b(11ab + 9) \end{aligned}
Therefore, the equivalent expression to the given algebraic expression is 9b(11ab+9) 9b(11ab + 9) . This matches choice 2.

Thus, the solution to the problem is 9b(11ab+9) 9b(11ab + 9) .

Answer

9b(11ab+9) 9b(11ab+9)

Exercise #8

Which of the expressions is equivalent to the expression?

6mn+3nm+9n2 6mn+\frac{3n}{m}+9n^2

Video Solution

Step-by-Step Solution

To solve the problem, we'll follow these steps:

  • Step 1: Identify the common factor in the expression 6mn+3nm+9n2 6mn+\frac{3n}{m}+9n^2 .
  • Step 2: Factor this common factor out from the expression.
  • Step 3: Compare the factored expression to the given choices.

Let's work through each step:

Step 1: Identify the Common Factor

Looking at the terms 6mn 6mn , 3nm\frac{3n}{m}, and 9n2 9n^2 , the common factor among them is clearly 3n 3n since:

  • 6mn 6mn can be divided by 3n 3n , giving 2m 2m .
  • 3nm \frac{3n}{m} can be divided by 3n 3n , giving 1m\frac{1}{m}.
  • 9n2 9n^2 can be divided by 3n 3n , giving 3n 3n .

Step 2: Factor Out the Common Factor

Factoring 3n 3n out of each term, we rewrite the expression:

6mn+3nm+9n2=3n(2m)+3n(1m)+3n(3n) 6mn + \frac{3n}{m} + 9n^2 = 3n(2m) + 3n\left(\frac{1}{m}\right) + 3n(3n) .

This simplifies to:

3n(2m+1m+3n) 3n(2m + \frac{1}{m} + 3n) .

Step 3: Compare with Choices

We compare our factored expression, 3n(2m+1m+3n) 3n(2m + \frac{1}{m} + 3n) , to the given choices. We find that Choice 1 matches our factored form.

Therefore, the expression 6mn+3nm+9n2 6mn+\frac{3n}{m}+9n^2 is equivalent to 3n(2m+1m+3n) 3n(2m+\frac{1}{m}+3n) .

Answer

3n(2m+1m+3n) 3n(2m+\frac{1}{m}+3n)

Exercise #9

Which of the expressions are equal to the expression?

2ab4bc 2ab-4bc

  1. 2b(a2c) 2b(a-2c)

  2. 2b(2c+a) 2b(-2c+a)

  3. 2(2bc+ab) 2(-2bc+ab)

  4. 2a(2bcb) 2a(2bc-b)

Video Solution

Step-by-Step Solution

Let's determine the equivalence of different expressions to 2ab4bc 2ab-4bc by factorization:

Step 1: Factor the given expression:

The expression 2ab4bc 2ab - 4bc has a common factor of 2b 2b .

Factor out 2b 2b , we obtain:

2ab4bcamp;=2b(a)2b(2c)amp;=2b(a2c) \begin{aligned} 2ab - 4bc &= 2b(a) - 2b(2c) \\ &= 2b(a - 2c) \end{aligned}

Step 2: Compare with each option:

  • Option 1: 2b(a2c) 2b(a-2c)

    • This is identical to the factorized form 2b(a2c) 2b(a-2c) , so it is equivalent.

  • Option 2: 2b(2c+a) 2b(-2c+a)

    • Although it appears reversed, 2c+a -2c + a is equivalent to a2c a - 2c , so it's equivalent.

  • Option 3: 2(2bc+ab) 2(-2bc+ab)

    • By rearranging:

      2(2bc+ab)amp;=2(ab2bc)amp;=2ab4bc \begin{aligned} 2(-2bc + ab) &= 2(ab - 2bc) \\ &= 2ab - 4bc \end{aligned}

    • It matches the original expression, thus is equivalent.

  • Option 4: 2a(2bcb) 2a(2bc-b)

    • Expanding:

      2a(2bcb)=4abc2ab 2a(2bc-b) = 4abc - 2ab

    • Does not match 2ab4bc 2ab - 4bc , so not equivalent.

Conclusion: The expressions equivalent to 2ab4bc 2ab - 4bc are Options 1, 2, and 3.

Therefore, the solution to the problem is 1,2,3 1,2,3 .

Answer

1,2,3 1,2,3

Exercise #10

Which of the expressions are equal to the expression?

12n+4836mn144m 12n+48-36mn-144m

  1. 12(13m)(n+4) 12(1-3m)(n+4)

  2. 3(412m)(n+4) 3(4-12m)(n+4)

  3. 12(3m+4)(n+1) 12(-3m+4)(n+1)

  4. 12(n+4)3m(12n+48) 12(n+4)-3m(12n+48)

Video Solution

Step-by-Step Solution

Let's start by factoring the given expression 12n+4836mn144m 12n + 48 - 36mn - 144m .

First, notice that:

  • In the first two terms: 12n+48 12n + 48 , we can factor out 12 12 , giving us 12(n+4) 12(n + 4) .

  • In the last two terms: 36mn144m -36mn - 144m , we can factor out 36m -36m , resulting in 36m(n+4) -36m(n + 4) .

Combining both factorizations, we can write the original expression as:

12(n+4)36m(n+4) 12(n + 4) - 36m(n + 4)

Now, we factor out the common term (n+4)(n + 4):

(1236m)(n+4) (12 - 36m)(n + 4)

Thus, the expression simplifies to:

12(13m)(n+4) 12(1 - 3m)(n + 4)

Now, let's verify which options match:

Option 1: 12(13m)(n+4) 12(1-3m)(n+4)
This directly matches our simplified expression, so it is a correct choice.

Option 2: 3(412m)(n+4) 3(4 - 12m)(n + 4)
Simplifying: Factoring 3 from 412m 4 - 12m gives 3×4(13m) 3 \times 4(1 - 3m) , which matches the expression 12(13m)(n+4) 12(1 - 3m)(n + 4) . So, this is also a correct choice.

Option 3: 12(3m+4)(n+1) 12(-3m + 4)(n + 1)
The factors do not align with our expression because (n+1)(n+1) is not factored from (n+4) (n + 4) .

Option 4: 12(n+4)3m(12n+48) 12(n + 4) - 3m(12n + 48)
Rewriting: 12(n+4)3m(12(n+4))=(1236m)(n+4) 12(n+4) - 3m(12(n+4)) = (12 - 36m)(n+4) which matches. Therefore, it is correct.

Hence, options 1, 2, and 4 are equivalent to the original expression.

The correct answer to the problem is

1,2,4 1, 2, 4

Answer

1,2,4 1,2,4

Exercise #11

Which of the following expressions have the same value?

  1. (6b+3)(2+a) (6b+3)(-2+a)

  2. (2b+1)(3a6) (2b+1)(3a-6)

  3. (a+3)(6b2) (a+3)(6b-2)

  4. 6ab+3a12b6 6ab+3a-12b-6

Video Solution

Step-by-Step Solution

To solve this problem, we need to systematically expand and simplify each expression given in the problem statement:

  • Expression 1: (6b+3)(2+a) (6b+3)(-2+a)

    Expand using the distributive property:

    =6b(2)+6b(a)+3(2)+3(a) = 6b(-2) + 6b(a) + 3(-2) + 3(a)

    =12b+6ab6+3a = -12b + 6ab - 6 + 3a

    Reorder terms: 6ab+3a12b6 6ab + 3a - 12b - 6

  • Expression 2: (2b+1)(3a6) (2b+1)(3a-6)

    Expand using the distributive property:

    =2b(3a)+2b(6)+1(3a)+1(6) = 2b(3a) + 2b(-6) + 1(3a) + 1(-6)

    =6ab12b+3a6 = 6ab - 12b + 3a - 6

    This simplifies directly to 6ab+3a12b6 6ab + 3a - 12b - 6

  • Expression 3: (a+3)(6b2) (a+3)(6b-2)

    Expand using the distributive property:

    =a(6b)+a(2)+3(6b)+3(2) = a(6b) + a(-2) + 3(6b) + 3(-2)

    =6ab2a+18b6 = 6ab - 2a + 18b - 6

    This results in 6ab2a+18b6 6ab - 2a + 18b - 6 , clearly different from the others

  • Expression 4: 6ab+3a12b6 6ab+3a-12b-6

    This is already simplified and the same as the results of expressions 1 and 2.

Upon comparing the simplified expressions, expressions 1, 2, and 4 have the same value: 6ab+3a12b6 6ab + 3a - 12b - 6 . Expression 3 differs with 6ab2a+18b6 6ab - 2a + 18b - 6 .

Thus, the expressions with the same value are 1, 2, and 4.

Therefore, the correct answer is choice 4: 1,2,4 1,2,4 .

Answer

1,2,4 1,2,4

Exercise #12

Which of the expressions are equal to the expression?

14mn+21m2 \frac{14m^{}}{n}+21m^2

  1. 7m(2n+3m) 7m(\frac{2}{n}+3m)

  2. 7(2nm+3m2) 7(\frac{2}{nm}+3m^2)

  3. mn(14+3m) \frac{m}{n}(14+3m)

  4. 7mn(2+3mn) 7\frac{m}{n}(2+3mn)

Video Solution

Step-by-Step Solution

To solve this problem, we'll simplify and compare the given expressions. Let's begin by factoring the original expression:

Factor the original expression 14mn+21m2 \frac{14m}{n} + 21m^2 :

Notice both terms contain a factor of 7m 7m :

14mn+21m2=7m(2n+3m) \frac{14m}{n} + 21m^2 = 7m\left(\frac{2}{n} + 3m\right)

Now, let's examine each given choice:

  • Choice 1: 7m(2n+3m) 7m\left(\frac{2}{n} + 3m\right) matches the factored version we derived from the original, so they are equivalent.
  • Choice 2: Simplifying 7(2nm+3m2) 7\left(\frac{2}{nm} + 3m^2\right) :

    =72nm+73m2=14nm+21m2 = 7 \cdot \frac{2}{nm} + 7 \cdot 3m^2 = \frac{14}{nm} + 21m^2 ; note this is different from the original expression since we require 14mn \frac{14m}{n} in the first term.

  • Choice 3: Simplicity is tested for mn(14+3m) \frac{m}{n}(14 + 3m) :

    =14mn+3m2n = \frac{14m}{n} + \frac{3m^2}{n} ; for equivalence, recall we needed complete 14mn+21m2 \frac{14m}{n} + 21m^2 , not 3m2n\frac{3m^2}{n}

  • Choice 4: Examine 7mn(2+3mn) 7\frac{m}{n}(2 + 3mn) :

    Bifurcation gives =14mn+21m2 = \frac{14m}{n} + 21m^2 ; multipliers yield the original expression accurately.

Thus, the answer is clearly seen that both choices 1 and 4 are equivalent to the original expression:

The correct choices are 1 and 4.

Answer

1,4 1,4

Exercise #13

Which of the expressions are equal to the expression?

36tr18rt2 \frac{36t}{r}-18rt^2

  1. 18(2trrt2) 18(\frac{2t}{r}-rt^2)

  2. 18r(2tr2t2) 18r(\frac{2t}{r^2}-t^2)

  3. 18(rt+2tr) -18(rt+\frac{2t}{r})

  4. 6t(6tr3rt) 6t(\frac{6t}{r}-3rt)

Video Solution

Step-by-Step Solution

To solve this problem, let's simplify and factor the given expression:

Original expression: 36tr18rt2 \frac{36t}{r} - 18rt^2

Step 1: Factor out the greatest common divisor, which is 18:

18(2trrt2) 18 \left( \frac{2t}{r} - rt^2 \right)

This matches the structure of choice 1.

Step 2: Substitute this back into the given options and simplify.

Option 1: 18(2trrt2) 18\left( \frac{2t}{r} - rt^2 \right)

This is identical to the factored form of the original.

Option 2: 18r(2tr2t2) 18r\left( \frac{2t}{r^2} - t^2 \right)

Expand and simplify:

=18r2tr218rt2 = 18r \cdot \frac{2t}{r^2} - 18r \cdot t^2

=36tr18rt2 = \frac{36t}{r} - 18rt^2

This matches the original expression.

Option 3: 18(rt+2tr) -18(rt + \frac{2t}{r})

Expand and simplify:

=18rt36tr = -18rt - \frac{36t}{r}

This does not match the original expression.

Option 4: 6t(6tr3rt) 6t\left( \frac{6t}{r} - 3rt \right)

Expand and simplify:

=6t6tr6t3rt = 6t \cdot \frac{6t}{r} - 6t \cdot 3rt

=36t2r18rt2 = \frac{36t^2}{r} - 18rt^2

This does not match the original expression.

Therefore, the correct options that are equivalent to the given expression are 1 and 2.

Answer

1,2 1,2

Exercise #14

Which of the expressions are equal to the expression?

6x2+8xy 6x^2+8xy

  1. 2(3x24xy) -2(-3x^2-4xy)

  2. 2(3x2+4xy) 2(3x^2+4xy)

  3. 2x(3x+4y) 2x(3x+4y)

  4. 2y(3x2y+4x) 2y(\frac{3x^2}{y}+4x)

Video Solution

Answer

All expressions are equal