Question

PLEASE HELP ME WITH ALGEBRA PROBLEMS PLEASEEEE

Answer 1

1. The remainder is 1.

2. The remainder is 0.

3. The required number is 3.

Explanation:

1.

The dividend is
`3x^3-2x^2+x-1`.

The divisor is (x-1).

The coefficient of dividend are 3,-2, 1 and -1. The divisor is (x-1), so we multiply dropped or added value by 1.

Write all the coefficients in top row. Drop down first coefficient as it is.

Multiply it by 1 and write the result in middle row, then add them.

Use synthetic method as shown in the below figure.

Bottom row represent the coefficients of quotient and remainder.

`3x^3-2x^2+x-1=3x^2+x+2+(1)/(x-1)`

The remainder is 1.

2.

If the divisor is a factor of given polynomial, then the polynomial is completely divides by divisor and the remainder is 0.

3.

The divisor is (x-3).

The divisor in the form of (x-c). The we multiply dropped or added value by c.

Since the divisor in the form of (x-3). The we multiply dropped or added value by 3.

Answer 2

Explanation:

In the first part the given polynomial is
`3x^(3)-2x^(2)+x-1`, we have to divide this polynomial by (x-1) using synthetic division.

Here are the steps of synthetic division

Step 1: Set up the synthetic division table in which the coefficients are written in descending power of x.

Step 2: Drop down the leading coefficient i.e the coefficient of highest power of x to the bottom row.

Step 3: Multiply a by the value just written on the bottom row and write this value just down of the number written next of leading coefficient. Here, a is 1(taken from divisor)

Step 4: Add the column made in step 3. Repeat these steps until the last column reached.

In the second part,

It is given that divisor polynomial is the factor of given polynomial then we have to tell the remainder which is 0 because if divisor polynomial is the factor the it will completely divide the given polynomial which gives remainder 0.

In the third part,

Given when one uses the synthetic division to divide by (x-3) then what number is used to multiply each term that is dropped down or added.

As explained the steps of synthetic division in first part

x-3=0 ⇒ x=3 which is a

In step 3, Multiply a by the value just written on the bottom row and write this value just down of the number written next of leading coefficient. So, 3 is multiplied.

As explained in Step 2: Drop down the leading coefficient i.e the coefficient of highest power of x to the bottom row.

The columns which is formed by above steps are added.