Find and simplify h(x)/g(x); hint: you will need to factor h(x)
h(x)= 16x^2 - 14x - 36
g(x) = 2x - 4
so, did you follow the hint? Apparently not.
h(x) = 2(x-2)(8x+9)
g(x) = 2(x-2)
now finish it off
To find and simplify h(x)/g(x), we need to perform the following steps:
Step 1: Factor h(x)
To factorize h(x) = 16x^2 - 14x - 36, we need to look for two numbers that multiply to -576 (the product of the leading coefficient and the constant term) and add up to the coefficient of the middle term (-14).
The pairs of factors of -576 are:
-1, 576
-2, 288
-3, 192
-4, 144
-6, 96
-8, 72
-9, 64
-12, 48
-16, 36
-18, 32
-24, 24
After trying out these pairs, we find that -18 and 32 add up to give us -14. Thus, we can factor h(x) as:
h(x) = (4x - 6)(4x + 6)
Step 2: Simplify h(x)/g(x)
Now that we have factored h(x) as (4x - 6)(4x + 6), we can rewrite h(x)/g(x) as:
h(x)/g(x) = [(4x - 6)(4x + 6)] / (2x - 4)
To simplify this fraction, we can cancel out common factors:
h(x)/g(x) = [(4x - 6)(4x + 6)] / (2x - 4)
= [(2(2x - 3))(2(2x + 3))] / (2(x - 2))
= [(2 * 2 * (x - 2))(2(x + 3))] / (2(x - 2))
= (2 * 2 * (x - 2)(x + 3)) / (2(x - 2))
Now, we can cancel out the common factors, which are (x - 2) and 2:
h(x)/g(x) = (2 * 2 * (x - 2)(x + 3)) / (2(x - 2))
= (2 * (x + 3)) / 1
= 2(x + 3)
Therefore, the simplified form of h(x)/g(x) is 2(x + 3).
To find and simplify h(x)/g(x), we first need to factor the numerator, h(x).
The expression h(x) = 16x^2 - 14x - 36 can be factored as follows:
h(x) = (4x + 6)(4x - 6)
Now, we can rewrite our expression as (4x + 6)(4x - 6) / g(x), where g(x) = 2x - 4.
Next, we can simplify the expression by canceling any common factors between the numerator and denominator.
The common factor between (4x + 6) and g(x) = 2(x - 2).
So, we can write the expression as:
(4x - 6) / 2(x - 2)
Further simplifying, we can cancel the 2s:
(2x - 3) / (x - 2)
Therefore, h(x)/g(x) simplifies to (2x - 3) / (x - 2).