isiah is dividing $2x^{3}-x^{2}+2x + 5$ by $x + 1$ using a division table. his work is shown here. what is…
isiah is dividing $2x^{3}-x^{2}+2x + 5$ by $x + 1$ using a division table. his work is shown here. what is the value of a? -5 5 1x 5x
Answer
Explanation:
Step1: Use polynomial long - division or synthetic division concept
We can use polynomial long - division. When we divide a polynomial (P(x)=2x^{3}-x^{2}+2x + 5) by (x + 1), we can also think in terms of the distributive property. Let's assume (2x^{3}-x^{2}+2x + 5=(x + 1)(ax^{2}+bx + c)+r). First, we know that when we multiply ((x + 1)) by (ax^{2}), the leading term of the product is (ax^{3}). Since the leading term of (2x^{3}-x^{2}+2x + 5) is (2x^{3}), then (a = 2). ((x + 1)(2x^{2}+bx + c)=2x^{3}+bx^{2}+cx+2x^{2}+bx + c=2x^{3}+(b + 2)x^{2}+(b + c)x + c). Comparing the coefficients of (x^{2}): (-1=b + 2), so (b=-3). Then ((x + 1)(2x^{2}-3x + c)=2x^{3}-3x^{2}+cx+2x^{2}-3x + c=2x^{3}-x^{2}+(c - 3)x + c). Comparing the coefficients of (x): (2=c - 3), so (c = 5).
Step2: Analyze the division process
When we divide (2x^{3}-x^{2}+2x + 5) by (x + 1) using a division - table method (which is related to the distributive property of multiplication), the constant term in the quotient when we consider the division of the constant part of the dividend by the linear factor is (5).
Answer:
5