Answer :
We are given the coefficients of the polynomial:
[tex]$$4, \quad 6, \quad -3$$[/tex]
and we want to divide the polynomial
[tex]$$4x^2 + 6x - 3$$[/tex]
by [tex]$(x - 1)$[/tex]. The synthetic division process for such a division is as follows:
1. Write down the coefficients:
[tex]$$4, \quad 6, \quad -3.$$[/tex]
We use the zero of the divisor [tex]$(x-1)$[/tex], which is [tex]$x = 1$[/tex], as our synthetic divisor.
2. Bring down the first coefficient directly:
[tex]$$\text{New coefficient} = 4.$$[/tex]
3. Multiply the number just brought down by the synthetic divisor:
[tex]$$4 \times 1 = 4.$$[/tex]
Then add this product to the next coefficient:
[tex]$$6 + 4 = 10.$$[/tex]
So, our new second number is [tex]$10$[/tex].
4. Multiply this new number by the synthetic divisor:
[tex]$$10 \times 1 = 10.$$[/tex]
Add this product to the constant term:
[tex]$$-3 + 10 = 7.$$[/tex]
This final number is the remainder.
Thus, the remainder after dividing [tex]$4x^2 + 6x - 3$[/tex] by [tex]$(x-1)$[/tex] is
[tex]$$\boxed{7}.$$[/tex]
[tex]$$4, \quad 6, \quad -3$$[/tex]
and we want to divide the polynomial
[tex]$$4x^2 + 6x - 3$$[/tex]
by [tex]$(x - 1)$[/tex]. The synthetic division process for such a division is as follows:
1. Write down the coefficients:
[tex]$$4, \quad 6, \quad -3.$$[/tex]
We use the zero of the divisor [tex]$(x-1)$[/tex], which is [tex]$x = 1$[/tex], as our synthetic divisor.
2. Bring down the first coefficient directly:
[tex]$$\text{New coefficient} = 4.$$[/tex]
3. Multiply the number just brought down by the synthetic divisor:
[tex]$$4 \times 1 = 4.$$[/tex]
Then add this product to the next coefficient:
[tex]$$6 + 4 = 10.$$[/tex]
So, our new second number is [tex]$10$[/tex].
4. Multiply this new number by the synthetic divisor:
[tex]$$10 \times 1 = 10.$$[/tex]
Add this product to the constant term:
[tex]$$-3 + 10 = 7.$$[/tex]
This final number is the remainder.
Thus, the remainder after dividing [tex]$4x^2 + 6x - 3$[/tex] by [tex]$(x-1)$[/tex] is
[tex]$$\boxed{7}.$$[/tex]
