fill in the missing values below one at a time to find the quotient when $x^{3}+11x^{2}+23x - 8$ is divided…

fill in the missing values below one at a time to find the quotient when $x^{3}+11x^{2}+23x - 8$ is divided by $x + 8$.

fill in the missing values below one at a time to find the quotient when $x^{3}+11x^{2}+23x - 8$ is divided by $x + 8$.

Answer

Explanation:

Step1: Multiply (x) and (3x^{2})

We know that in the area - model of polynomial division (similar to the area - model of multiplication). If we have a term (x\times) (a term in the quotient). The first term of the dividend is (x^{3}) and we know that (x\times x^{2}=x^{3}). The second term of the dividend is (11x^{2}), and we already have (8x^{2}) from (8\times x^{2}). Let the second term of the quotient be (ax). Then (x\times ax+8x^{2}=11x^{2}).

We use the formula for the product of monomials (x\times ax = ax^{2}). So, (ax^{2}+8x^{2}=11x^{2}). By the property of equality for polynomials (equating the coefficients of like - terms), (a + 8=11), which gives (a = 3).

The term we are looking for is (x\times3x) (since the second term of the quotient is (3x)). Using the rule of multiplying monomials (x\times3x=3x^{2})

Answer:

(3x^{2})