Factor completely. Remember to look first for a common factor. Check by multiplying. If a polynomial is prime state this.
a^2-2ab-3b^2
what if it were
x^2 - 2 x - 3 ??
(x-3)(x+1)
now try
To factor the given polynomial, let's begin by looking for a common factor. In this case, the greatest common factor (GCF) of the terms is 1, so there isn't a common factor to factor out.
Next, we can proceed with factoring the polynomial using the method of factoring by grouping or quadratic factoring.
The given polynomial is: a^2 - 2ab - 3b^2
To factor this quadratic expression, we need to find two binomials in the form of (ax + by) that multiply together to give us the original quadratic expression.
To do this, we look for two numbers that multiply together to give us the constant term (-3 in this case) and add up to give the coefficient of the middle term (-2ab in this case).
The factors of -3 are (-1, 3) and (1, -3). After trying them out, we find that -1 and 3 meet our requirements - they add up to -2 and multiply to -3.
Now, we can rewrite the middle term (-2ab) as -ab + 3ab:
a^2 - ab + 3ab - 3b^2
We can now group the terms and factor by grouping:
(a^2 - ab) + (3ab - 3b^2)
We can factor out the common terms from each group:
a(a - b) + 3b(a - b)
Now we have a common binomial factor of (a - b):
(a - b)(a + 3b)
So, the fully factored form of the given polynomial is (a - b)(a + 3b).
To check if we factored correctly, we can multiply the factors (a - b)(a + 3b) using the distributive property:
(a - b)(a + 3b) = a(a) + a(3b) - b(a) - b(3b) = a^2 + 3ab - ab - 3b^2
Simplifying the expression gives us the original polynomial, a^2 - 2ab - 3b^2.
Therefore, the factored form is correct.
Hope this helps!