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Answer :
Certainly! When you're dividing a polynomial [tex]\( f(x) \)[/tex] by a linear factor using synthetic division, evaluating the polynomial at a certain value is connected to the remainder of this division.
In the situation shown, Cristoble is using synthetic division to divide the polynomial [tex]\( f(x) \)[/tex] by [tex]\( x + 3 \)[/tex]. The key point is that synthetic division not only helps in dividing polynomials but also in evaluating the polynomial at specific points. Specifically, the remainder you obtain in synthetic division is the value of the polynomial evaluated at the value that makes the divisor zero.
Here's a step-by-step explanation of what Cristoble has done:
1. Synthetic Division Setup:
The divisor is [tex]\( x + 3 \)[/tex], which means we will evaluate [tex]\( f(x) \)[/tex] at [tex]\( -3 \)[/tex]. This is because if we set [tex]\( x + 3 = 0 \)[/tex], we solve for [tex]\( x = -3 \)[/tex].
2. Coefficients from Polynomial:
Look at the top row of the table: [tex]\( 2, -5, 3 \)[/tex]. These numbers represent the coefficients of the polynomial you are dividing. So, the polynomial [tex]\( f(x) \)[/tex] might look something like [tex]\( 2x^2 - 5x + 3 \)[/tex].
3. Synthetic Division Process:
- Start by bringing down the first coefficient, 2.
- Multiply the root ([tex]\(-3\)[/tex]) by this first coefficient (2), yielding -6, and write it under the second coefficient (-5).
- Add the value under the second coefficient: [tex]\(-5 + (-6) = -11\)[/tex].
- Multiply [tex]\(-11\)[/tex] by [tex]\(-3\)[/tex] to get 33, write it under the third coefficient (3).
- Finally, add this to the last coefficient: [tex]\(3 + 33 = 36\)[/tex].
4. Remainder and Value of [tex]\( f(-3) \)[/tex]:
The final number we get after all additions is 36. This is the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 3 \)[/tex], and it also equals [tex]\( f(-3) \)[/tex].
Therefore, the value of [tex]\( f(-3) \)[/tex] is 36.
In the situation shown, Cristoble is using synthetic division to divide the polynomial [tex]\( f(x) \)[/tex] by [tex]\( x + 3 \)[/tex]. The key point is that synthetic division not only helps in dividing polynomials but also in evaluating the polynomial at specific points. Specifically, the remainder you obtain in synthetic division is the value of the polynomial evaluated at the value that makes the divisor zero.
Here's a step-by-step explanation of what Cristoble has done:
1. Synthetic Division Setup:
The divisor is [tex]\( x + 3 \)[/tex], which means we will evaluate [tex]\( f(x) \)[/tex] at [tex]\( -3 \)[/tex]. This is because if we set [tex]\( x + 3 = 0 \)[/tex], we solve for [tex]\( x = -3 \)[/tex].
2. Coefficients from Polynomial:
Look at the top row of the table: [tex]\( 2, -5, 3 \)[/tex]. These numbers represent the coefficients of the polynomial you are dividing. So, the polynomial [tex]\( f(x) \)[/tex] might look something like [tex]\( 2x^2 - 5x + 3 \)[/tex].
3. Synthetic Division Process:
- Start by bringing down the first coefficient, 2.
- Multiply the root ([tex]\(-3\)[/tex]) by this first coefficient (2), yielding -6, and write it under the second coefficient (-5).
- Add the value under the second coefficient: [tex]\(-5 + (-6) = -11\)[/tex].
- Multiply [tex]\(-11\)[/tex] by [tex]\(-3\)[/tex] to get 33, write it under the third coefficient (3).
- Finally, add this to the last coefficient: [tex]\(3 + 33 = 36\)[/tex].
4. Remainder and Value of [tex]\( f(-3) \)[/tex]:
The final number we get after all additions is 36. This is the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 3 \)[/tex], and it also equals [tex]\( f(-3) \)[/tex].
Therefore, the value of [tex]\( f(-3) \)[/tex] is 36.
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