Certainly! Let's step through each question.
### Question 7: Value of [tex]\( f(2) \)[/tex] for [tex]\( f(x) = 3x^2 - 4 \)[/tex]
1. Given Function:
[tex]\( f(x) = 3x^2 - 4 \)[/tex]
2. Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[
f(2) = 3 \cdot (2)^2 - 4
\][/tex]
3. Calculate [tex]\( (2)^2 \)[/tex]:
[tex]\[
(2)^2 = 4
\][/tex]
4. Multiply by 3:
[tex]\[
3 \cdot 4 = 12
\][/tex]
5. Subtract 4:
[tex]\[
12 - 4 = 8
\][/tex]
6. Therefore, the value of [tex]\( f(2) \)[/tex] is:
[tex]\[
f(2) = 8
\][/tex]
### Question 8: Sum of the Polynomials [tex]\( 2x^3 - 4x^2 + 3x - 1 \)[/tex] and [tex]\( x^3 + 2x^2 - 5x + 3 \)[/tex]
1. Given Polynomials:
- First polynomial: [tex]\( 2x^3 - 4x^2 + 3x - 1 \)[/tex]
- Second polynomial: [tex]\( x^3 + 2x^2 - 5x + 3 \)[/tex]
2. Identify corresponding coefficients:
- First polynomial coefficients: [tex]\([2, -4, 3, -1]\)[/tex]
- Second polynomial coefficients: [tex]\([1, 2, -5, 3]\)[/tex]
3. Add corresponding coefficients:
[tex]\[
\begin{array}{c|c|c}
\text{Degree} & \text{First Polynomial} & \text{Second Polynomial} & \text{Sum} \\
\hline
x^3 & 2 & 1 & 2 + 1 = 3 \\
x^2 & -4 & 2 & -4 + 2 = -2 \\
x & 3 & -5 & 3 + (-5) = -2 \\
1 & -1 & 3 & -1 + 3 = 2 \\
\end{array}
\][/tex]
4. Sum of the polynomials:
[tex]\[
3x^3 - 2x^2 - 2x + 2
\][/tex]
5. Therefore, the resulting polynomial is:
[tex]\[
3x^3 - 2x^2 - 2x + 2
\][/tex]
These are the detailed step-by-step solutions:
- [tex]\( f(2) = 8 \)[/tex]
- The sum of the polynomials is [tex]\( 3x^3 - 2x^2 - 2x + 2 \)[/tex]
