Certainly! Let's solve each problem step-by-step.
### Question 5.3: Add the polynomials [tex]\( (3x - 7x^2 + 4) \)[/tex] and [tex]\( (3 + 2x - x^2) \)[/tex]
Step-by-Step Solution:
1. Write down the given polynomials:
[tex]\[
P1 = 3x - 7x^2 + 4
\][/tex]
[tex]\[
P2 = 3 + 2x - x^2
\][/tex]
2. Combine like terms:
- Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
-7x^2 - x^2 = -8x^2
\][/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\[
3x + 2x = 5x
\][/tex]
- Combine the constant terms:
[tex]\[
4 + 3 = 7
\][/tex]
3. Write the resulting polynomial:
[tex]\[
P1 + P2 = -8x^2 + 5x + 7
\][/tex]
Final Answer:
[tex]\[
-8x^2 + 5x + 7
\][/tex]
---
### Question 5.4: James and Mpho together have 127 stamps. James has 4 more than twice as many as Mpho. How many stamps do each of them have?
Step-by-Step Solution:
1. Define the variables:
Let [tex]\( m \)[/tex] be the number of stamps Mpho has.
2. Express James' number of stamps in terms of [tex]\( m \)[/tex]:
James' stamps = [tex]\( 2m + 4 \)[/tex]
3. Set up the equation based on the total number of stamps:
[tex]\[
m + (2m + 4) = 127
\][/tex]
4. Simplify the equation:
[tex]\[
3m + 4 = 127
\][/tex]
5. Solve for [tex]\( m \)[/tex]:
Subtract 4 from both sides:
[tex]\[
3m = 123
\][/tex]
Divide both sides by 3:
[tex]\[
m = 41
\][/tex]
So, Mpho has [tex]\( 41 \)[/tex] stamps.
6. Calculate the number of stamps James has:
[tex]\[
2m + 4 = 2(41) + 4 = 82 + 4 = 86
\][/tex]
Final Answer:
- Mpho has [tex]\( 41 \)[/tex] stamps.
- James has [tex]\( 86 \)[/tex] stamps.
