https://file.aiquickdraw.com/mathbot/file/1726506128824w04ifgzd.jpg

I see that the image contains three mathematical problems involving second-degree polynomial functions (quadratic functions). Here's what each problem asks:

1. **Problem 1**: "Déterminer les fonctions polynômes du second degré s'annulant en 1 et en -12."
   - You are asked to find the quadratic function that has roots at \( x = 1 \) and \( x = -12 \). 

2. **Problem 2**: "Déterminer la fonction polynôme du second degré s'annulant en -4 et 3 et telle que \( f(4) = 4 \)."
   - Here, the task is to find the quadratic function that has roots at \( x = -4 \) and \( x = 3 \), with an additional condition that \( f(4) = 4 \).

3. **Problem 3**: "Déterminer la fonction polynôme du second degré \( g \) de racines \( -1 \) et \( 1 \) et telle que \( g(0) = 1 \)."
   - In this case, you are to determine the quadratic function with roots at \( x = -1 \) and \( x = 1 \), and the function value \( g(0) = 1 \).

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Let me go through each of these problems step by step:

### 1. Problem 1:
We know the function has roots at \( x = 1 \) and \( x = -12 \), so the polynomial can be written as:
\[
f(x) = a(x - 1)(x + 12)
\]
where \( a \) is a constant to be determined if any extra conditions are given.

### 2. Problem 2:
The roots are \( x = -4 \) and \( x = 3 \), so the general form of the quadratic is:
\[
f(x) = a(x + 4)(x - 3)
\]
Using the condition \( f(4) = 4 \), we can substitute \( x = 4 \) into the equation to find \( a \).

### 3. Problem 3:
The roots are \( x = -1 \) and \( x = 1 \), so the general form of the quadratic is:
\[
g(x) = a(x + 1)(x - 1) = a(x^2 - 1)
\]
Given that \( g(0) = 1 \), substitute \( x = 0 \) and solve for \( a \).

Would you like me to solve these problems step by step, starting with the first one, or provide detailed solutions for all three?

1. Déterminer les fonctions polynômes du second degré s'annulant en 1 et en -12. 2. Déterminer la fonction polynôme du second degré s'annulant en -4 et 3 et telle que f(4) = 4. 3. Déterminer la fonction polynôme du second degré g de racines -1 et 1 et telle que g(0) = 1.

Learn how to solve quadratic polynomial functions by determining roots and applying conditions such as f(4) = 4 and g(0) = 1. This tutorial focuses on solving polynomials with specific roots and additional functional constraints, suitable for high school algebra students.

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