Answer :
Sure! Let's use synthetic division to determine if [tex]\( (x + 5) \)[/tex] is a factor of [tex]\( 4x^3 + 19x^2 - 4x + 5 \)[/tex].
### Step-by-step Solution:
1. Set up for Synthetic Division:
- Since we are checking if [tex]\( (x + 5) \)[/tex] is a factor, we use [tex]\( x = -5 \)[/tex] in synthetic division.
- Write down the coefficients of the polynomial: [tex]\( 4, 19, -4, \)[/tex] and [tex]\( 5 \)[/tex].
2. Perform Synthetic Division:
- Bring down the first coefficient: [tex]\( 4 \)[/tex].
- Multiply [tex]\( 4 \)[/tex] by [tex]\(-5\)[/tex] (the number from [tex]\( x = -5 \)[/tex]), which gives [tex]\(-20\)[/tex].
- Add [tex]\(-20\)[/tex] to the next coefficient [tex]\( 19 \)[/tex], resulting in [tex]\(-1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\( 5 \)[/tex].
- Add [tex]\( 5 \)[/tex] to the next coefficient [tex]\(-4\)[/tex], resulting in [tex]\( 1 \)[/tex].
- Multiply [tex]\( 1 \)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-5\)[/tex].
- Add [tex]\(-5\)[/tex] to the last coefficient [tex]\( 5 \)[/tex], resulting in [tex]\( 0 \)[/tex].
3. Identify Quotient and Remainder:
- The quotient from our division is [tex]\( 4x^2 - x + 1 \)[/tex].
- The remainder is [tex]\( 0 \)[/tex].
4. Determine if [tex]\( (x + 5) \)[/tex] is a Factor:
- Since the remainder is [tex]\( 0 \)[/tex], [tex]\( (x + 5) \)[/tex] is indeed a factor of the polynomial [tex]\( 4x^3 + 19x^2 - 4x + 5 \)[/tex].
So, the quotient is [tex]\( 4x^2 - x + 1 \)[/tex], the remainder is 0, and yes, [tex]\( (x + 5) \)[/tex] is a factor of the polynomial.
### Step-by-step Solution:
1. Set up for Synthetic Division:
- Since we are checking if [tex]\( (x + 5) \)[/tex] is a factor, we use [tex]\( x = -5 \)[/tex] in synthetic division.
- Write down the coefficients of the polynomial: [tex]\( 4, 19, -4, \)[/tex] and [tex]\( 5 \)[/tex].
2. Perform Synthetic Division:
- Bring down the first coefficient: [tex]\( 4 \)[/tex].
- Multiply [tex]\( 4 \)[/tex] by [tex]\(-5\)[/tex] (the number from [tex]\( x = -5 \)[/tex]), which gives [tex]\(-20\)[/tex].
- Add [tex]\(-20\)[/tex] to the next coefficient [tex]\( 19 \)[/tex], resulting in [tex]\(-1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\( 5 \)[/tex].
- Add [tex]\( 5 \)[/tex] to the next coefficient [tex]\(-4\)[/tex], resulting in [tex]\( 1 \)[/tex].
- Multiply [tex]\( 1 \)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-5\)[/tex].
- Add [tex]\(-5\)[/tex] to the last coefficient [tex]\( 5 \)[/tex], resulting in [tex]\( 0 \)[/tex].
3. Identify Quotient and Remainder:
- The quotient from our division is [tex]\( 4x^2 - x + 1 \)[/tex].
- The remainder is [tex]\( 0 \)[/tex].
4. Determine if [tex]\( (x + 5) \)[/tex] is a Factor:
- Since the remainder is [tex]\( 0 \)[/tex], [tex]\( (x + 5) \)[/tex] is indeed a factor of the polynomial [tex]\( 4x^3 + 19x^2 - 4x + 5 \)[/tex].
So, the quotient is [tex]\( 4x^2 - x + 1 \)[/tex], the remainder is 0, and yes, [tex]\( (x + 5) \)[/tex] is a factor of the polynomial.
