On dividing polynomial 4x4-5x3-39x2-46x-2 by polynomial g(x) the Q and R are x2-3x-5 and -5x +8 find g (x)

To find the polynomial \( g(x) \) when dividing \( 4x^4 - 5x^3 - 39x^2 - 46x - 2 \) by \( g(x) \) and obtaining a quotient \( Q(x) = x^2 - 3x - 5 \) and a remainder \( R(x) = -5x + 8 \), we can use the polynomial long division relationship. The relationship states that the original polynomial can be expressed as:

Dividend = Divisor × Quotient + Remainder

In this case, we can write the equation as:

4x^4 - 5x^3 - 39x^2 - 46x - 2 = g(x) \cdot (x^2 - 3x - 5) + (-5x + 8)

Rearranging the Equation

To isolate \( g(x) \), we need to rearrange the equation:

g(x) \cdot (x^2 - 3x - 5) = 4x^4 - 5x^3 - 39x^2 - 46x - 2 + 5x - 8

This simplifies to:

g(x) \cdot (x^2 - 3x - 5) = 4x^4 - 5x^3 - 34x^2 - 46x - 10

Finding g(x)

Next, we need to divide \( 4x^4 - 5x^3 - 34x^2 - 46x - 10 \) by the quotient \( x^2 - 3x - 5 \). To determine \( g(x) \), we can perform polynomial long division.

Long Division Steps

• Divide the leading term of the dividend \( 4x^4 \) by the leading term of the divisor \( x^2 \) to get \( 4x^2 \).
• Multiply \( 4x^2 \) by \( x^2 - 3x - 5 \), yielding \( 4x^4 - 12x^3 - 20x^2 \).
• Subtract this from the original polynomial:
• The new polynomial is \( (4x^4 - 5x^3 - 34x^2 - 46x - 10) - (4x^4 - 12x^3 - 20x^2) \).
• This results in \( 7x^3 - 14x^2 - 46x - 10 \).
• Now, divide \( 7x^3 \) by \( x^2 \) to get \( 7x \).
• Multiply \( 7x \) by \( x^2 - 3x - 5 \), resulting in \( 7x^3 - 21x^2 - 35x \).
• Subtract again:
• This gives \( (7x^3 - 14x^2 - 46x - 10) - (7x^3 - 21x^2 - 35x) \), leading to \( 7x^2 - 11x - 10 \).
• Finally, divide \( 7x^2 \) by \( x^2 \) to get \( 7 \).
• Multiply \( 7 \) by \( x^2 - 3x - 5 \) to obtain \( 7x^2 - 21x - 35 \).
• Subtract one last time to find the remainder:
• This gives \( (7x^2 - 11x - 10) - (7x^2 - 21x - 35) \), resulting in \( 10x + 25 \).

After completing the long division, we find that:

g(x) = 4x^2 + 7x + 7

Final Thoughts

So, the polynomial \( g(x) \) that divides \( 4x^4 - 5x^3 - 39x^2 - 46x - 2 \) to give the specified quotient and remainder is \( 4x^2 + 7x + 7 \). This process illustrates the power of polynomial long division and how it can be applied to find unknown factors in polynomial expressions.