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7. Write the value of [tex]\( f(2) \)[/tex] if [tex]\( f(x) = 3x^2 - 4 \)[/tex].

8. Find the sum of the polynomials [tex]\( 2x^3 - 4x^2 + 3x - 1 \)[/tex] and [tex]\( x^3 + 2x^2 - 5x + 3 \)[/tex].



Answer :

GinnyAnswer
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]

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