Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Certainly! Let's go through the factorization process for each expression step-by-step:
### 1. [tex]\( xy - 2y + 3x + 6 \)[/tex]
Group terms to factor by grouping:
[tex]\[ (xy - 2y) + (3x + 6) \][/tex]
Factor out common factors from each group:
[tex]\[ y(x - 2) + 3(x - 2) \][/tex]
Now, factor out the common binomial factor:
[tex]\[ (y + 3)(x - 2) \][/tex]
So, the factorized form is:
[tex]\[ (y + 3)(x - 2) \][/tex]
### 2. [tex]\( P q\left(r^2 + 1\right) - r\left(p^2 + q^2\right) \)[/tex]
This expression does not easily simplify through simple factorization methods. However, we can rewrite it for clarity:
[tex]\[ pqr^2 + pq - rp^2 - rq^2 \][/tex]
Here, there's no common continuous factor or group to factor by. It remains in its current form.
### 3. [tex]\( 2x^2 + 7x + 6 \)[/tex]
To factor this quadratic expression, we find two numbers that multiply to [tex]\( 2 \times 6 = 12 \)[/tex] and add to [tex]\( 7 \)[/tex]:
[tex]\[ 2x^2 + 7x + 6 = 2x^2 + 3x + 4x + 6 \][/tex]
Group terms for factorization:
[tex]\[ = (2x^2 + 3x) + (4x + 6) \][/tex]
Factor out the common factors:
[tex]\[ = x(2x + 3) + 2(2x + 3) \][/tex]
Factor out the common binomial factor:
[tex]\[ = (2x + 3)(x + 2) \][/tex]
So, the factorized form is:
[tex]\[ (2x + 3)(x + 2) \][/tex]
### 4. [tex]\( 12 \frac{a^2}{b^2} + \frac{a}{b} - 20 \)[/tex]
First, rewrite the expression to make it easier to manage:
[tex]\[ 12 \frac{a^2}{b^2} + \frac{a}{b} - 20 = \frac{12a^2 + ab - 20b^2}{b^2} \][/tex]
This quadratic expression in the numerator can be factorized by finding two numbers that multiply to [tex]\( 12 \times (-20) = -240 \)[/tex] and add to [tex]\( 1 \)[/tex]:
[tex]\[ \frac{(4a - 5b)(3a + 4b)}{b^2} \][/tex]
So, the factorized form is:
[tex]\[ \frac{(4a - 5b)(3a + 4b)}{b^2} \][/tex]
### 5. [tex]\( \frac{x^2}{y^2} - 2 - \frac{8y^2}{z^2} \)[/tex]
This expression involves different variables and terms. It can be simplified but not easily factored in a typical sense.
### Additional Factorization Problems:
#### i. [tex]\( 25a^2b^2 - 1 \)[/tex]
This is a difference of squares:
[tex]\[ (5ab)^2 - 1^2 = (5ab - 1)(5ab + 1) \][/tex]
So, the factorized form is:
[tex]\[ (5ab - 1)(5ab + 1) \][/tex]
#### ii. [tex]\( 4 - (m - n)^2 \)[/tex]
This is another difference of squares:
[tex]\[ 4 - (m - n)^2 = 2^2 - (m - n)^2 = (2 + (m - n))(2 - (m - n)) \][/tex]
Simplify:
[tex]\[ (2 + m - n)(2 - m + n) \][/tex]
So, the factorized form is:
[tex]\[ (2 + m - n)(2 - m + n) \][/tex]
#### iii. [tex]\( a^2 - b^2 - 45 - 4b + 4\)[/tex]
Rearrange terms to recognize another potential difference of squares or groupwise simplification:
[tex]\[ a^2 - b^2 + 4 - 45 - 4b \][/tex]
This does not straightforwardly simplify further without context.
#### iv. [tex]\( 16a^4 - 4a^2 - 4a - 1 \)[/tex]
Group terms appropriately:
[tex]\[ 16a^4 - (4a^2 + 4a + 1) = (4a^2)^2 - (2a + 1)^2 \][/tex]
Recognize the difference of squares:
[tex]\[ (4a^2 - (2a + 1))(4a^2 + (2a + 1)) \][/tex]
So, the factorized form is:
[tex]\[ (4a^2 - 2a - 1)(4a^2 + 2a + 1) \][/tex]
#### v. [tex]\( x^3 + 6x + 5 - 4y - y^2 \)[/tex]
This expression involves multiple variables and terms. Factoring directly isn't straightforward without context or further grouping.
Each factorization has specific techniques; some are easily grouped, while others require recognizing patterns or more advanced algebraic methods.
### 1. [tex]\( xy - 2y + 3x + 6 \)[/tex]
Group terms to factor by grouping:
[tex]\[ (xy - 2y) + (3x + 6) \][/tex]
Factor out common factors from each group:
[tex]\[ y(x - 2) + 3(x - 2) \][/tex]
Now, factor out the common binomial factor:
[tex]\[ (y + 3)(x - 2) \][/tex]
So, the factorized form is:
[tex]\[ (y + 3)(x - 2) \][/tex]
### 2. [tex]\( P q\left(r^2 + 1\right) - r\left(p^2 + q^2\right) \)[/tex]
This expression does not easily simplify through simple factorization methods. However, we can rewrite it for clarity:
[tex]\[ pqr^2 + pq - rp^2 - rq^2 \][/tex]
Here, there's no common continuous factor or group to factor by. It remains in its current form.
### 3. [tex]\( 2x^2 + 7x + 6 \)[/tex]
To factor this quadratic expression, we find two numbers that multiply to [tex]\( 2 \times 6 = 12 \)[/tex] and add to [tex]\( 7 \)[/tex]:
[tex]\[ 2x^2 + 7x + 6 = 2x^2 + 3x + 4x + 6 \][/tex]
Group terms for factorization:
[tex]\[ = (2x^2 + 3x) + (4x + 6) \][/tex]
Factor out the common factors:
[tex]\[ = x(2x + 3) + 2(2x + 3) \][/tex]
Factor out the common binomial factor:
[tex]\[ = (2x + 3)(x + 2) \][/tex]
So, the factorized form is:
[tex]\[ (2x + 3)(x + 2) \][/tex]
### 4. [tex]\( 12 \frac{a^2}{b^2} + \frac{a}{b} - 20 \)[/tex]
First, rewrite the expression to make it easier to manage:
[tex]\[ 12 \frac{a^2}{b^2} + \frac{a}{b} - 20 = \frac{12a^2 + ab - 20b^2}{b^2} \][/tex]
This quadratic expression in the numerator can be factorized by finding two numbers that multiply to [tex]\( 12 \times (-20) = -240 \)[/tex] and add to [tex]\( 1 \)[/tex]:
[tex]\[ \frac{(4a - 5b)(3a + 4b)}{b^2} \][/tex]
So, the factorized form is:
[tex]\[ \frac{(4a - 5b)(3a + 4b)}{b^2} \][/tex]
### 5. [tex]\( \frac{x^2}{y^2} - 2 - \frac{8y^2}{z^2} \)[/tex]
This expression involves different variables and terms. It can be simplified but not easily factored in a typical sense.
### Additional Factorization Problems:
#### i. [tex]\( 25a^2b^2 - 1 \)[/tex]
This is a difference of squares:
[tex]\[ (5ab)^2 - 1^2 = (5ab - 1)(5ab + 1) \][/tex]
So, the factorized form is:
[tex]\[ (5ab - 1)(5ab + 1) \][/tex]
#### ii. [tex]\( 4 - (m - n)^2 \)[/tex]
This is another difference of squares:
[tex]\[ 4 - (m - n)^2 = 2^2 - (m - n)^2 = (2 + (m - n))(2 - (m - n)) \][/tex]
Simplify:
[tex]\[ (2 + m - n)(2 - m + n) \][/tex]
So, the factorized form is:
[tex]\[ (2 + m - n)(2 - m + n) \][/tex]
#### iii. [tex]\( a^2 - b^2 - 45 - 4b + 4\)[/tex]
Rearrange terms to recognize another potential difference of squares or groupwise simplification:
[tex]\[ a^2 - b^2 + 4 - 45 - 4b \][/tex]
This does not straightforwardly simplify further without context.
#### iv. [tex]\( 16a^4 - 4a^2 - 4a - 1 \)[/tex]
Group terms appropriately:
[tex]\[ 16a^4 - (4a^2 + 4a + 1) = (4a^2)^2 - (2a + 1)^2 \][/tex]
Recognize the difference of squares:
[tex]\[ (4a^2 - (2a + 1))(4a^2 + (2a + 1)) \][/tex]
So, the factorized form is:
[tex]\[ (4a^2 - 2a - 1)(4a^2 + 2a + 1) \][/tex]
#### v. [tex]\( x^3 + 6x + 5 - 4y - y^2 \)[/tex]
This expression involves multiple variables and terms. Factoring directly isn't straightforward without context or further grouping.
Each factorization has specific techniques; some are easily grouped, while others require recognizing patterns or more advanced algebraic methods.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
