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Answer :
To divide the polynomial [tex]\( f(x) = x^3 - 4x^2 - 47x + 210 \)[/tex] by [tex]\( x - 5 \)[/tex] using synthetic division and then find all the zeros of [tex]\( f(x) \)[/tex], follow these steps:
1. Set up synthetic division:
First, write down the coefficients of the polynomial [tex]\( f(x) = x^3 - 4x^2 - 47x + 210 \)[/tex]. They are [tex]\( 1, -4, -47, \)[/tex] and [tex]\( 210 \)[/tex].
Since we are dividing by [tex]\( x - 5 \)[/tex], use the zero of this binomial, which is [tex]\( 5 \)[/tex].
2. Perform synthetic division:
- Write [tex]\( 5 \)[/tex] on the left side and the coefficients [tex]\( 1, -4, -47, 210 \)[/tex] on the right side.
- Bring down the first coefficient, [tex]\( 1 \)[/tex].
- Multiply [tex]\( 5 \)[/tex] by the value you just brought down ([tex]\( 1 \)[/tex]) and write the result, [tex]\( 5 \)[/tex], under the next coefficient, [tex]\(-4\)[/tex].
- Add [tex]\(-4\)[/tex] and [tex]\(5\)[/tex] to get [tex]\(1\)[/tex]. Write this below the line.
- Repeat this process: multiply [tex]\(5\)[/tex] by [tex]\(1\)[/tex] to get [tex]\(5\)[/tex], add to [tex]\(-47\)[/tex] to get [tex]\(-42\)[/tex].
- Finally, multiply [tex]\(5\)[/tex] by [tex]\(-42\)[/tex] to get [tex]\(-210\)[/tex], add to [tex]\(210\)[/tex] to get [tex]\(0\)[/tex].
Here is the division set up:
```
5 | 1 -4 -47 210
| 5 5 -210
---------------------
1 1 -42 0
```
The bottom row gives us the coefficients of the quotient polynomial and shows the remainder. The quotient is [tex]\( x^2 + x - 42 \)[/tex] with a remainder of [tex]\( 0 \)[/tex].
3. Find the zeros of the polynomial [tex]\( f(x) \)[/tex]:
Since the remainder is [tex]\( 0 \)[/tex], [tex]\( x - 5 \)[/tex] is a factor. Therefore, [tex]\( 5 \)[/tex] is a zero of the polynomial.
Now, use the quotient [tex]\( x^2 + x - 42 \)[/tex] to find the other zeros. To do this, set the quadratic [tex]\( x^2 + x - 42 = 0 \)[/tex].
Solve the quadratic equation by factoring:
- Find two numbers that multiply to [tex]\(-42\)[/tex] and add to [tex]\(1\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(-7\)[/tex].
- So, [tex]\( x^2 + x - 42 = (x + 7)(x - 6) \)[/tex].
Hence, the solutions are [tex]\( x = -7 \)[/tex] and [tex]\( x = 6 \)[/tex].
4. Conclusion:
The zeros of the polynomial [tex]\( f(x) = x^3 - 4x^2 - 47x + 210 \)[/tex] are [tex]\( -7, 5, \)[/tex] and [tex]\( 6 \)[/tex].
1. Set up synthetic division:
First, write down the coefficients of the polynomial [tex]\( f(x) = x^3 - 4x^2 - 47x + 210 \)[/tex]. They are [tex]\( 1, -4, -47, \)[/tex] and [tex]\( 210 \)[/tex].
Since we are dividing by [tex]\( x - 5 \)[/tex], use the zero of this binomial, which is [tex]\( 5 \)[/tex].
2. Perform synthetic division:
- Write [tex]\( 5 \)[/tex] on the left side and the coefficients [tex]\( 1, -4, -47, 210 \)[/tex] on the right side.
- Bring down the first coefficient, [tex]\( 1 \)[/tex].
- Multiply [tex]\( 5 \)[/tex] by the value you just brought down ([tex]\( 1 \)[/tex]) and write the result, [tex]\( 5 \)[/tex], under the next coefficient, [tex]\(-4\)[/tex].
- Add [tex]\(-4\)[/tex] and [tex]\(5\)[/tex] to get [tex]\(1\)[/tex]. Write this below the line.
- Repeat this process: multiply [tex]\(5\)[/tex] by [tex]\(1\)[/tex] to get [tex]\(5\)[/tex], add to [tex]\(-47\)[/tex] to get [tex]\(-42\)[/tex].
- Finally, multiply [tex]\(5\)[/tex] by [tex]\(-42\)[/tex] to get [tex]\(-210\)[/tex], add to [tex]\(210\)[/tex] to get [tex]\(0\)[/tex].
Here is the division set up:
```
5 | 1 -4 -47 210
| 5 5 -210
---------------------
1 1 -42 0
```
The bottom row gives us the coefficients of the quotient polynomial and shows the remainder. The quotient is [tex]\( x^2 + x - 42 \)[/tex] with a remainder of [tex]\( 0 \)[/tex].
3. Find the zeros of the polynomial [tex]\( f(x) \)[/tex]:
Since the remainder is [tex]\( 0 \)[/tex], [tex]\( x - 5 \)[/tex] is a factor. Therefore, [tex]\( 5 \)[/tex] is a zero of the polynomial.
Now, use the quotient [tex]\( x^2 + x - 42 \)[/tex] to find the other zeros. To do this, set the quadratic [tex]\( x^2 + x - 42 = 0 \)[/tex].
Solve the quadratic equation by factoring:
- Find two numbers that multiply to [tex]\(-42\)[/tex] and add to [tex]\(1\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(-7\)[/tex].
- So, [tex]\( x^2 + x - 42 = (x + 7)(x - 6) \)[/tex].
Hence, the solutions are [tex]\( x = -7 \)[/tex] and [tex]\( x = 6 \)[/tex].
4. Conclusion:
The zeros of the polynomial [tex]\( f(x) = x^3 - 4x^2 - 47x + 210 \)[/tex] are [tex]\( -7, 5, \)[/tex] and [tex]\( 6 \)[/tex].
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