show that (2x-3) is a factor of 2x³+3x²-5x-6 hence factorize (2x³+3x²-5x-6).
Answers
Answer:
Step-by-step explanation:
Factoring a cubic polynomial is not very easy. I suggest trying out a combination of p/q approach for polynomials and factor and/or remainder theorem.
p/q, where p are the possible integer factors of the constant (in your case that is 6), and q are the possible integer factors of the coefficient of the leading term (in your case that is 2). Thus
p = 1, 2, 3, 6
q = 1, 2
p/q = 1, 1/2, 2, 3, 3/2, 6
which may be either positive or negative. p/q lists down the possible factors of your polynomial, so this will help you greatly factor out any polynomial.
Using factor and/or remainder theorem, you can determine whether the numbers you have are roots (possible factors) of a polynomial. You apply it by substituting a number for the x's, and if the answer comes out a 0, then that number is a root. Let's try that now.
2(1)^3 + 3(1)^2 - 5(1) - 6 = -6
2(-1)^3 + 3(-1)^2 - 5(-1) - 6 = 0
Since it zeroes at -1, then -1 is a root, which makes (x - (-1)) or (x+1) a factor of the cubic.
Thus, we just need to figure out what would be the remaining polynomial, after we take out (x+1) from it. In short,
2x^3 + 3x^2 - 5x - 6 = (x + 1)(...)
To determine this, we need to use either synthetic division or long division. I prefer synthetic division.
2 3 -5 -6 | -1 (because this is the root)
-2 -1 6
2 1 -6 0
Thus, the remaining polynomial's coefficients are 2, 1 and -6. The remaining polynomial is a polynomial which is a degree lower than the starting polynomial. Because the starting polynomial is cubic (degree 3), the remaining polynomial is quadratic (degree 2)
2x^3 + 3x^2 - 5x - 6 = (x + 1)(2x^2 + x - 6)
Since the remaining polynomial is quadratic, we can just simply factor out using traditional factoring.
2x^2 + x - 6 = (2x - 3)(x + 2)
Thus, the factored form of the cubic is
2x^3 + 3x^2 - 5x - 6 = (x + 1)(2x - 3)(x + 2)
You may expand the factors by multiplying to check.
Hope this helps!!!!!