Solve Polynomial Division: -5a + 4b = x^2+1 Rem -A-2

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In summary, the remainder of p(x) divided by x^2+1 is –5a + 4b. When divided by x+1, the constant term of the resulting equation is 4b-a+1. To solve for 8ab, you need to set 4b-a+1 equal to 0 and solve for the value of x.
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The remainder of \(\displaystyle p(x)=x^3+ax^2+4bx-1\) divided by \(\displaystyle x^2+1\) is –5a + 4b. If the remainder of p(x) divided by x + 1 is –a – 2, the value of 8ab is ...
A. \(\displaystyle -\frac34\)
B. \(\displaystyle -\frac12\)
C. 0
D. 1
E. 3

Dividing p(x) by \(\displaystyle x^2+1\) by \(\displaystyle x^2+1\) with –5a + 4b as the remainder using long division, I got (4bx – 1) – ((a – 1)x + a – 1) = –5a + 4b, thus (4b – a + 1)x + a = –5a + 4b. Does this mean that 4b – a + 1 = 0 since the right hand doesn't have an x term? Or do I need to look for the value of x first? I'm at a loss here.
 
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yes. 4b – a + 1 = 0 as there is no x term.
you need to divide by x+1 and get the constant term (putting x = -1 shall do also)
this shall give 2 equations and you need to solve and proceed further;
 

FAQ: Solve Polynomial Division: -5a + 4b = x^2+1 Rem -A-2

What is polynomial division?

Polynomial division is the process of dividing one polynomial expression by another. It is similar to long division in arithmetic, where the quotient and remainder are calculated.

How do you solve polynomial division?

To solve polynomial division, you need to follow the steps of long division. First, arrange the terms of the dividend and divisor in descending order of their exponents. Then, divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. Multiply this term by the entire divisor and subtract it from the dividend. Repeat this process until there are no more terms left in the dividend, and the remainder is smaller than the divisor.

What is the remainder in polynomial division?

The remainder in polynomial division is the leftover term after the division process is complete. It represents the amount that cannot be divided evenly by the divisor.

How do you solve the given polynomial division problem?

To solve the given polynomial division problem, you need to first rearrange the terms to put them in descending order of their exponents. Then, follow the steps of long division to divide the first term of the dividend by the first term of the divisor, multiply it by the entire divisor, and subtract it from the dividend. Continue this process until there are no more terms left in the dividend, and the remainder is smaller than the divisor.

What is the significance of solving polynomial division?

Solving polynomial division is essential in simplifying polynomial expressions and solving equations involving polynomials. It also helps in finding the roots of polynomial equations and graphing polynomial functions.

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