I wanted to know how you would know that -4x^2 + 2x +90 / x-5 would be able to factor down to (x-5)(-4x-18)/x-5
By looking at -4x^2 + 2x +90 / x-5 , I would never think that it could be factored down to (x-5)(-4x-18)/x-5
How should I approach it so I know that it could be factored down to (x-5)(-4x-18)/x-5?
First you take out common factors.
-4x^2 + 2x +90 = -2(2x²-x-45)
Is it possible to find a, b such that ab=-45, 2a+b=-1 or a+2b=-1?
Look at the factors of the constant term, namely 45=5*9. Since 2*5-9=1, we can try the factors 5 and 9 in different ways to get:
(2x+9)(x-5)
rather hard to explain in detail here.
this page
http://www.recitfga.qc.ca/english/activities/sitsat-2006/jean-foster/0-3.htm
gives a reasonable explanation.
I became suspicious when I saw the simple factor x-5 in the denominator, and had a sneaking suspicion that it could also be in the top.
so I subbed in x=5 into the top and sure enough, I got a zero for the result.
(recall the factor theorem, which says that if f(a) = 0 then x-a is a factor)
so I know that
-4x^2 + 2x +90
= (x - 5)(?x ± ?)
now a bit of logic,
what multiplied by x would give me -4x^2 ? , clearly -4x
what multiplied by -5 would give me +90 ? clearly -18
so
-4x^2 + 2x +90 = (x-5)(-4x-18)
BTW, the answer can be taken further by dividing out the x-5 and simplifying the top
(x-5)(-4x-18)/x-5
= -2(2x + 9), x cannot be equal to 5
To determine if the expression -4x^2 + 2x + 90 / x - 5 can be factored, you can use the factoring rules and procedures. Follow these steps:
Step 1: Identify the expression as a polynomial. In this case, it is a quadratic polynomial.
Step 2: Check if the polynomial can be factored further by looking for common factors among the terms. In this polynomial, there are no common factors shared by all the terms.
Step 3: Examine the quadratic term coefficient (in front of the x^2 term) and the constant term (the term without an x) of the polynomial. In this case, the coefficient of the quadratic term is -4 and the constant term is 90.
Step 4: Look for two numbers that multiply to give the product of the quadratic term coefficient and constant term and add up to the coefficient of the linear term (the term with an x). In this case, you need to find two numbers that multiply to give -4 * 90 = -360 and add up to 2.
Step 5: Use these numbers to split the linear term. Rewrite the linear term (-2x) as the sum of two terms that use the numbers found in the previous step. In this case, you need to find two numbers that add up to 2 and multiply to give -360. These numbers are -18 and -20. So, you rewrite 2x as -18x - 20x.
Step 6: Group the terms and factor by grouping. Break the polynomial into two groups by grouping the first two terms together and the last two terms together. Factor out the greatest common factor from each group and then factor out any common factors that may be present between the two groups.
Group 1: -4x^2 + (-18x)
Group 2: 2x + (-20)
Step 7: Factor out the greatest common factor from each group. The greatest common factor in Group 1 is -2x, and in Group 2, it is 2.
Group 1: -2x(2x + 9)
Group 2: 2(x - 10)
Step 8: Check if there are any common factors between the two groups. Here, you can see that (2x + 9) is present in both groups, so you can factor it out.
Final factored form: -2x(2x + 9) + 2(x - 10)
Step 9: Simplify and cancel out like terms. In this case, you can simplify further by canceling out the common factor of (2x + 9).
Final factored form: (2x + 9)(-2x + 20)
Now you can see that the expression -4x^2 + 2x + 90 / x - 5 can be factored down to (x - 5)(-4x - 18) / (x - 5).