Sure! Let's solve each part of the given question step-by-step:
### Part (1)
We need to evaluate the polynomial [tex]\( x^5 - 2x^4 - 3x^2 \)[/tex] at [tex]\( x = \frac{1}{2} \)[/tex].
1. Substitute [tex]\( x = \frac{1}{2} \)[/tex] into the polynomial:
[tex]\[
\left(\frac{1}{2}\right)^5 - 2 \left(\frac{1}{2}\right)^4 - 3 \left(\frac{1}{2}\right)^2
\][/tex]
2. Calculate each term:
- [tex]\( \left(\frac{1}{2}\right)^5 = \frac{1}{32} \)[/tex]
- [tex]\( 2 \left(\frac{1}{2}\right)^4 = 2 \cdot \frac{1}{16} = \frac{2}{16} = \frac{1}{8} \)[/tex]
- [tex]\( 3 \left(\frac{1}{2}\right)^2 = 3 \cdot \frac{1}{4} = \frac{3}{4} \)[/tex]
3. Combine the results:
[tex]\[
\frac{1}{32} - \frac{1}{8} - \frac{3}{4}
\][/tex]
4. Convert all fractions to have a common denominator of 32:
[tex]\[
\frac{1}{32} - \frac{4}{32} - \frac{24}{32}
\][/tex]
5. Perform the subtraction:
[tex]\[
\frac{1}{32} - \frac{28}{32} = -\frac{27}{32}
\][/tex]
Thus, the value of [tex]\( x^5 - 2x^4 - 3x^2 \)[/tex] at [tex]\( x = \frac{1}{2} \)[/tex] is approximately [tex]\(-0.84375\)[/tex].
### Part (3)
We need to evaluate the expression [tex]\( 20t + 4.9t^2 \)[/tex] at [tex]\( t = 10 \)[/tex].
1. Substitute [tex]\( t = 10 \)[/tex] into the expression:
[tex]\[
20(10) + 4.9(10)^2
\][/tex]
2. Calculate each term:
- [tex]\( 20(10) = 200 \)[/tex]
- [tex]\( 4.9(10)^2 = 4.9(100) = 490 \)[/tex]
3. Combine the results:
[tex]\[
200 + 490 = 690
\][/tex]
Thus, the value of [tex]\( 20t + 4.9t^2 \)[/tex] at [tex]\( t = 10 \)[/tex] is [tex]\( 690 \)[/tex].
### Part (9)
We need to evaluate the polynomial [tex]\( 2x^4 + 5x^3 - x + 6 \)[/tex] at [tex]\( x = 2 \)[/tex].
1. Substitute [tex]\( x = 2 \)[/tex] into the polynomial:
[tex]\[
2(2)^4 + 5(2)^3 - 2 + 6
\][/tex]
2. Calculate each term:
- [tex]\( 2(2)^4 = 2(16) = 32 \)[/tex]
- [tex]\( 5(2)^3 = 5(8) = 40 \)[/tex]
- [tex]\( -2 \)[/tex]
- [tex]\( +6 \)[/tex]
3. Combine the results:
[tex]\[
32 + 40 - 2 + 6 = 76
\][/tex]
Thus, the value of [tex]\( 2x^4 + 5x^3 - x + 6 \)[/tex] at [tex]\( x = 2 \)[/tex] is [tex]\( 76 \)[/tex].
So, the evaluated answers for the given expressions are:
- [tex]\( x^5 - 2x^4 - 3x^2 \)[/tex] at [tex]\( x = \frac{1}{2} \)[/tex] is [tex]\(-0.84375\)[/tex].
- [tex]\( 20t + 4.9t^2 \)[/tex] at [tex]\( t = 10 \)[/tex] is [tex]\( 690 \)[/tex].
- [tex]\( 2x^4 + 5x^3 - x + 6 \)[/tex] at [tex]\( x = 2 \)[/tex] is [tex]\( 76 \)[/tex].
