Rewrite the middle term as the sum of two terms and then factor completely.
x2 + 19x + 90
Factor completely. ax – ay + x2 – xy
Rewrite the middle term as the sum of two terms and then factor by grouping.
x2 + 4x – 12
Complete the following statement. 6a2 – 5a + 1 = (3a – 1)( )
I will be happy to critique your thinking on this.
To rewrite the middle term as the sum of two terms and then factor completely, we use the method called "factoring by splitting the middle term".
1. Start with the expression: x² + 19x + 90.
2. Identify two numbers whose product is equal to the product of the first and last term, which in this case is 90. The numbers also need to add up to the middle term, which is 19.
3. The numbers that fit these criteria are 9 and 10. (9 * 10 = 90 and 9 + 10 = 19)
4. Rewrite the middle term using these numbers: x² + 9x + 10x + 90.
5. Factor the expression by grouping: (x² + 9x) + (10x + 90).
6. Factor the common terms separately: x(x + 9) + 10(x + 9).
7. Factor out the common binomial: (x + 10)(x + 9).
So the completely factored form of x² + 19x + 90 is (x + 10)(x + 9).
To factor completely the expression ax – ay + x² – xy, we rearrange the terms in a different form.
1. Start with the expression: ax - ay + x² - xy.
2. Group the terms: (ax - ay) + (x² - xy).
3. Factor out common factors from each binomial: a(x - y) + x(x - y).
4. Recognize that (x - y) is common to both terms.
5. Factor out the common binomial: (x - y)(a + x).
So the completely factored form of ax - ay + x² - xy is (x - y)(a + x).
To rewrite the middle term as the sum of two terms and then factor by grouping in the expression x² + 4x - 12:
1. Start with the expression: x² + 4x - 12.
2. Identify two numbers whose product is equal to the product of the first and last term, which in this case is -12. The numbers also need to add up to the middle term, which is 4.
3. The numbers that fit these criteria are 6 and -2. (6 * -2 = -12 and 6 + -2 = 4)
4. Rewrite the middle term using these numbers: x² + 6x - 2x - 12.
5. Factor the expression by grouping: (x² + 6x) + (-2x - 12).
6. Factor the common terms separately: x(x + 6) - 2(x + 6).
7. Factor out the common binomial: (x - 2)(x + 6).
So the completely factored form of x² + 4x - 12 is (x - 2)(x + 6).
To complete the statement 6a² - 5a + 1 = (3a - 1)( ):
1. Start with the expression: 6a² - 5a + 1.
2. Identify two numbers whose product is equal to the product of the first and last term, which in this case is 6. The numbers also need to add up to the middle term, which is -5.
3. The numbers that fit these criteria are -3 and -2. (-3 * -2 = 6 and -3 + -2 = -5)
4. Rewrite the middle term using these numbers: 6a² - 3a - 2a + 1.
5. Group the terms: (6a² - 3a) + (-2a + 1).
6. Factor out the common terms separately: 3a(2a - 1) - 1(2a - 1).
7. Factor out the common binomial: (3a - 1)(2a - 1).
So the completely factored form of 6a² - 5a + 1 is (3a - 1)(2a - 1).
If you have any other questions or need further clarification, feel free to ask!