Algebra 1: Factoring Difference Of SquaresFactor Each Expression Completely.1) $16n^2 - 25$2) $4k^2 - 9$3) $12k^2 - 27$4) $9n^2 + 16$5) $r^2 - 25$6) $18b^2 - 50$7) $80a^2 - 125$8)

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Introduction

In algebra, factoring is a process of expressing a polynomial as a product of simpler polynomials. Factoring difference of squares is a specific technique used to factor expressions that can be written in the form of a difference of two squares. In this article, we will explore the concept of factoring difference of squares and provide step-by-step solutions to various examples.

What is Factoring Difference of Squares?

Factoring difference of squares is a technique used to factor expressions that can be written in the form of a difference of two squares. The general form of a difference of squares is:

a^2 - b^2 = (a + b)(a - b)

where a and b are constants or variables. This technique is used to simplify complex expressions and make them easier to work with.

How to Factor Difference of Squares?

To factor a difference of squares, we need to identify the two perfect squares that can be subtracted to get the original expression. We can do this by looking for two numbers whose product is the constant term and whose sum is the coefficient of the middle term.

Step 1: Identify the two perfect squares

The first step is to identify the two perfect squares that can be subtracted to get the original expression. We can do this by looking for two numbers whose product is the constant term and whose sum is the coefficient of the middle term.

Step 2: Write the expression as a difference of squares

Once we have identified the two perfect squares, we can write the expression as a difference of squares. This will give us the factored form of the expression.

Step 3: Factor the expression

The final step is to factor the expression by multiplying the two binomials.

Examples

Now that we have learned how to factor difference of squares, let's practice with some examples.

Example 1: 16n2−2516n^2 - 25

To factor this expression, we need to identify the two perfect squares that can be subtracted to get the original expression. We can do this by looking for two numbers whose product is 25 and whose sum is 0.

The two perfect squares are 16 and 1, so we can write the expression as:

16n2−25=(4n)2−5216n^2 - 25 = (4n)^2 - 5^2

Now we can factor the expression by multiplying the two binomials:

(4n)2−52=(4n+5)(4n−5)(4n)^2 - 5^2 = (4n + 5)(4n - 5)

Example 2: 4k2−94k^2 - 9

To factor this expression, we need to identify the two perfect squares that can be subtracted to get the original expression. We can do this by looking for two numbers whose product is 9 and whose sum is 0.

The two perfect squares are 4 and 1, so we can write the expression as:

4k2−9=(2k)2−324k^2 - 9 = (2k)^2 - 3^2

Now we can factor the expression by multiplying the two binomials:

(2k)2−32=(2k+3)(2k−3)(2k)^2 - 3^2 = (2k + 3)(2k - 3)

Example 3: 12k2−2712k^2 - 27

To factor this expression, we need to identify the two perfect squares that can be subtracted to get the original expression. We can do this by looking for two numbers whose product is 27 and whose sum is 0.

The two perfect squares are 12 and 1, so we can write the expression as:

12k2−27=(2k)2−3212k^2 - 27 = (2k)^2 - 3^2

Now we can factor the expression by multiplying the two binomials:

(2k)2−32=(2k+3)(2k−3)(2k)^2 - 3^2 = (2k + 3)(2k - 3)

Example 4: 9n2+169n^2 + 16

To factor this expression, we need to identify the two perfect squares that can be added to get the original expression. We can do this by looking for two numbers whose product is 16 and whose sum is 0.

The two perfect squares are 9 and 1, so we can write the expression as:

9n2+16=(3n)2+429n^2 + 16 = (3n)^2 + 4^2

Now we can factor the expression by multiplying the two binomials:

(3n)2+42=(3n+4)(3n−4)(3n)^2 + 4^2 = (3n + 4)(3n - 4)

Example 5: r2−25r^2 - 25

To factor this expression, we need to identify the two perfect squares that can be subtracted to get the original expression. We can do this by looking for two numbers whose product is 25 and whose sum is 0.

The two perfect squares are 25 and 1, so we can write the expression as:

r2−25=(r)2−52r^2 - 25 = (r)^2 - 5^2

Now we can factor the expression by multiplying the two binomials:

(r)2−52=(r+5)(r−5)(r)^2 - 5^2 = (r + 5)(r - 5)

Example 6: 18b2−5018b^2 - 50

To factor this expression, we need to identify the two perfect squares that can be subtracted to get the original expression. We can do this by looking for two numbers whose product is 50 and whose sum is 0.

The two perfect squares are 18 and 1, so we can write the expression as:

18b2−50=(3b)2−5218b^2 - 50 = (3b)^2 - 5^2

Now we can factor the expression by multiplying the two binomials:

(3b)2−52=(3b+5)(3b−5)(3b)^2 - 5^2 = (3b + 5)(3b - 5)

Example 7: 80a2−12580a^2 - 125

To factor this expression, we need to identify the two perfect squares that can be subtracted to get the original expression. We can do this by looking for two numbers whose product is 125 and whose sum is 0.

The two perfect squares are 80 and 1, so we can write the expression as:

80a2−125=(4a)2−5280a^2 - 125 = (4a)^2 - 5^2

Now we can factor the expression by multiplying the two binomials:

(4a)2−52=(4a+5)(4a−5)(4a)^2 - 5^2 = (4a + 5)(4a - 5)

Conclusion

Factoring difference of squares is a powerful technique used to simplify complex expressions and make them easier to work with. By identifying the two perfect squares that can be subtracted to get the original expression, we can write the expression as a difference of squares and factor it by multiplying the two binomials. With practice and patience, you can master this technique and become proficient in factoring difference of squares.

Practice Problems

Try factoring the following expressions:

  1. 36x2−4936x^2 - 49
  2. 25y2−14425y^2 - 144
  3. 81z2−6481z^2 - 64
  4. 49m2+3649m^2 + 36
  5. 100n2−121100n^2 - 121

Answer Key

  1. (6x+7)(6x−7)(6x + 7)(6x - 7)
  2. (5y+12)(5y−12)(5y + 12)(5y - 12)
  3. (9z+8)(9z−8)(9z + 8)(9z - 8)
  4. (7m+6)(7m−6)(7m + 6)(7m - 6)
  5. (10n+11)(10n−11)(10n + 11)(10n - 11)
    Algebra 1: Factoring Difference of Squares - Q&A =====================================================

Introduction

In our previous article, we explored the concept of factoring difference of squares and provided step-by-step solutions to various examples. In this article, we will answer some frequently asked questions about factoring difference of squares.

Q&A

Q: What is the difference of squares formula?

A: The difference of squares formula is:

a^2 - b^2 = (a + b)(a - b)

where a and b are constants or variables.

Q: How do I identify the two perfect squares in a difference of squares expression?

A: To identify the two perfect squares, look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term.

Q: Can I factor a difference of squares expression if it has a negative sign?

A: Yes, you can factor a difference of squares expression even if it has a negative sign. For example:

-25 = (-5)^2 - 5^2

Q: Can I factor a difference of squares expression if it has a fraction?

A: Yes, you can factor a difference of squares expression even if it has a fraction. For example:

(1/2)^2 - (1/3)^2

Q: How do I factor a difference of squares expression with a variable in the middle term?

A: To factor a difference of squares expression with a variable in the middle term, you need to identify the two perfect squares that can be subtracted to get the original expression. For example:

x^2 - 4y^2

Q: Can I factor a difference of squares expression if it has a coefficient on the variable?

A: Yes, you can factor a difference of squares expression even if it has a coefficient on the variable. For example:

3x^2 - 4y^2

Q: How do I factor a difference of squares expression with a negative coefficient on the variable?

A: To factor a difference of squares expression with a negative coefficient on the variable, you need to identify the two perfect squares that can be subtracted to get the original expression. For example:

-3x^2 + 4y^2

Q: Can I factor a difference of squares expression if it has a coefficient on the constant term?

A: Yes, you can factor a difference of squares expression even if it has a coefficient on the constant term. For example:

2x^2 - 9

Q: How do I factor a difference of squares expression with a negative coefficient on the constant term?

A: To factor a difference of squares expression with a negative coefficient on the constant term, you need to identify the two perfect squares that can be subtracted to get the original expression. For example:

-2x^2 + 9

Q: Can I factor a difference of squares expression if it has a coefficient on both the variable and the constant term?

A: Yes, you can factor a difference of squares expression even if it has a coefficient on both the variable and the constant term. For example:

3x^2 - 4

Q: How do I factor a difference of squares expression with a negative coefficient on both the variable and the constant term?

A: To factor a difference of squares expression with a negative coefficient on both the variable and the constant term, you need to identify the two perfect squares that can be subtracted to get the original expression. For example:

-3x^2 + 4

Conclusion

Factoring difference of squares is a powerful technique used to simplify complex expressions and make them easier to work with. By identifying the two perfect squares that can be subtracted to get the original expression, we can write the expression as a difference of squares and factor it by multiplying the two binomials. With practice and patience, you can master this technique and become proficient in factoring difference of squares.

Practice Problems

Try factoring the following expressions:

  1. 36x2−4936x^2 - 49
  2. 25y2−14425y^2 - 144
  3. 81z2−6481z^2 - 64
  4. 49m2+3649m^2 + 36
  5. 100n2−121100n^2 - 121

Answer Key

  1. (6x+7)(6x−7)(6x + 7)(6x - 7)
  2. (5y+12)(5y−12)(5y + 12)(5y - 12)
  3. (9z+8)(9z−8)(9z + 8)(9z - 8)
  4. (7m+6)(7m−6)(7m + 6)(7m - 6)
  5. (10n+11)(10n−11)(10n + 11)(10n - 11)