Factor:
A.) 2x^2-9x-18
B.) 14x^2-55x-36
C.) x^3+3x^2-x-3
A. (x-6)(2x+3)
B. (2x-9)(7x+4)
C. (x-1)(x+1)(x+3)
You must have learned a method which can be used for quadratics
One method which seems to be taught in many locations is the method of decomposition.
1. Multiply the first and last numbers:
2(-18) = -36
2. look for two factor of -36 which have a sum of -9 , (the middle term coefficient)
3.
-36
= -12(3)
3. now split the middle term of -9x into -12x + 3x
2x^2 - 12x + 3x - 18
partially factor it, that is, find a common facto for the first two terms and a common factor for the last two terms.
=2x(x-6) + 3(x-6)
You now must have a common factor, or else you made and error.
= (x-6)(2x+3)
A lot of students appear to like it, even though I have my own personal way which is much faster but rather hard to explain in this kind of forum.
Try the second one the same way, you should get the answer that Jai has given you.
The third one is similar to step #2 from above
To factor a quadratic expression, such as the ones given, we need to find two binomial expressions that, when multiplied together, will result in the original expression.
A.) To factor 2x^2-9x-18:
We are looking for two binomial expressions in the form (px + q) and (rx + s) where the product of the leading coefficients (p * r) is equal to 2, and the product of the constant terms (q * s) is equal to -18. Additionally, the sum of the cross product terms ((q * r) + (p * s)) should be equal to the coefficient of the linear term (-9x).
Let's try different combinations of numbers that satisfy the above conditions:
- Factors of 2: 1, 2
- Factors of -18: -1, 1, -2, 2, -3, 3, -6, 6, -9, 9, -18, 18
After examining the possible combinations of factors, we find that the factors of 2x^2-9x-18 can be written as:
(2x + 3)(x - 6)
B.) To factor 14x^2-55x-36:
Following the same process, we need to find factors of 14 and -36 that add up to -55 (coefficient of the linear term).
- Factors of 14: 1, 2, 7, 14
- Factors of -36: -1, 1, -2, 2, -3, 3, -4, 4, -6, 6, -9, 9, -12, 12, -18, 18, -36, 36
After reviewing the possible combinations, we can factor 14x^2-55x-36 as:
(7x - 9)(2x - 4)
C.) To factor x^3+3x^2-x-3:
Factoring a cubic expression is slightly more complicated than a quadratic expression.
The rational root theorem can help identify potential rational roots. In this case, the constant term (-3) is the potential numerator, and the leading coefficient (1) is the potential denominator.
We can test the potential roots by substituting them into the expression to see if they result in a zero value. After testing a few possible values, we find that x = 1 is one of the roots.
To proceed with the factoring, we can use synthetic division or long division to divide the original expression by (x - 1). Performing synthetic division, we get:
1 │ 1 3 -1 -3
- 1 4 3
_______________
1 4 3 0
The result gives us a quadratic expression, 1x^2 + 4x + 3. We can now factor this quadratic expression by finding two binomial expressions that multiply together to give us the quadratic and then combine the factored expression with (x - 1).
This quadratic expression can be factored as:
(x + 1)(x + 3)
Therefore, the factored form of x^3+3x^2-x-3 is:
(x - 1)(x + 1)(x + 3)