Question

Find r(x) where f(x) = -4x² - 14x - 35 - 2x - 7 and g(x) = -4x² - 4x²x + 7.

Answer 1

r(x) = (-7 - 2x) / (-8) where f(x) = -4x² - 14x - 35 - 2x - 7 and g(x) = -4x² - 4x²x + 7.

Step-by-step explanation:

To find the remainder when polynomial f(x) is divided by polynomial g(x), we use the formula:

r(x) = f(x) mod g(x)

First, let's find the quotient and remainder when f(x) is divided by g(x):

(-4x² - 14x - 35 - 2x - 7) mod (-4x² - 4x³ - 7)

To do this, we need to find the largest power of x that is common to both polynomials. In this case, it is x². Then, we divide the leading term of f(x) by the leading term of g(x):

(-4/-4) * (-4x²) = 1 * (-4x²)

Next, we subtract the product of the quotient and the divisor from the dividend:

(-4x² - 14x - 35 - 2x - 7) - (1 * (-4x²)) = (-20x - 35 - 7)

Now, we bring down the next term of f(x):

(-20x - 35 - 7) - (0 * (-4x³)) = (-20x - 35 - 7)

We repeat this process until we get a remainder that is not divisible by the divisor:

(-20x - 35 - 7) - (0 * (-4)) = (-7 - 20x) / (-8)

Therefore, r(x) is equal to (-7 - 20x) / (-8). This means that when we divide f(x) by g(x), there will be a remainder of (-7 - 20x) when x is substituted into the quotient and remainder expressions.

Answer 2

The expression for
`\( r(x) \) is \( r(x) = f(x) / g(x) \)`, which simplifies to
`\( r(x) = (-4x² - 14x - 35 - 2x - 7)/(-4x² - 4x²x + 7) \).`

Step-by-step explanation:

In the given problem, we are tasked with finding the rational function
`\( r(x) \) where \( f(x) = -4x² - 14x - 35 - 2x - 7 \) and \( g(x) = -4x² - 4x²x + 7 \)`

To find
`\( r(x) \), we divide \( f(x) \) by \( g(x) \): \( r(x) = (f(x))/(g(x)) \).`

Now, substitute the expressions for ( f(x) ) and ( g(x) ) into the division:

`\[ r(x) = (-4x² - 14x - 35 - 2x - 7)/(-4x² - 4x²x + 7) \]`

Simplify the numerator and denominator separately by combining like terms. After simplification, the expression for ( r(x) ) will be obtained.

It's crucial to note that we should exclude any values of ( x ) that make the denominator equal to zero, as division by zero is undefined. So, the final expression for( r(x) ) will be the rational function that represents the given conditions.

In summary, finding ( r(x) ) involves dividing ( f(x) ) by ( g(x) ), simplifying the resulting expression, and being mindful of any restrictions on the domain due to potential division by zero.