Question

Rewrite the expression: (x³+10x²+13x+39) / (+10x²+2x+1) in the form q(x)+ b(x)/r(x):

Answer 1

To rewrite the given expression, polynomial long division must be performed. The division will result in a quotient, which is expected to be a first-degree polynomial, and a remainder, which will be a polynomial of degree less than the divisor. This process involves dividing terms of similar degree and carefully handling any fractional coefficients.

Step-by-step explanation:

To rewrite the expression (x³+10x²+13x+39) / (+10x²+2x+1) in the form q(x) + b(x)/r(x) where q(x) is the quotient, b(x) is the remainder, and r(x) is the divisor, you would perform polynomial long division.

However, in this case, since the divisor is a second-degree polynomial and the dividend is a third-degree polynomial, we expect that the quotient q(x) will be a first-degree polynomial and the remainder b(x) will be of a degree less than the divisor. This means our division will yield a linear polynomial plus a fractional remainder.

To start the long division process, you would divide the highest-degree term of the dividend (x³) by the highest-degree term in the divisor (10x²) to get the first term of the quotient (1/10x). You would then multiply the entire divisor by this term and subtract it from the dividend, bringing down the next term of the dividend and repeating the process.

This process continues until all terms of the original dividend have been handled, at which point any term left that is of lower degree than the divisor becomes the remainder, b(x). The final answer will be this first-degree polynomial plus the remainder over the original divisor.

Note that if any coefficients are decimals or fractions, they need to be handled with care during the division to ensure accuracy.

Answer 2

The rewritten expression is q(x) + b(x)/r(x), where q(x) = x + 3 and b(x) = -27x + 12, and r(x) = 10x² + 2x + 1.

Step-by-step explanation:

To rewrite the given expression in the form q(x) + b(x)/r(x), we perform polynomial long division. The dividend is x³ + 10x² + 13x + 39, and the divisor is 10x² + 2x + 1. Dividing x into 10x² gives us x, so our quotient term, q(x), is x. Multiplying the entire divisor by x gives x³ + 2x² + x, which we subtract from the dividend to get the remainder, -27x + 38.

Now, we bring down the next term, 13x, and repeat the process. Dividing x into -27x gives us -27, so we add -27x - 27 to our quotient. Multiplying the entire divisor by -27 gives -27x² - 54x - 27, which we subtract from the remaining dividend, 13x + 65, resulting in a new remainder of 103.

Finally, we bring down the last term, 39, and divide x into 103, giving us +103. So, our final quotient, q(x), is x - 27, and the remainder, b(x), is 103. The divisor, r(x), remains 10x² + 2x + 1.

Therefore, the expression (x³ + 10x² + 13x + 39) / (10x² + 2x + 1) can be written as q(x) + b(x)/r(x), where q(x) = x - 27, b(x) = 103, and r(x) = 10x² + 2x + 1.