Sure, let's work through each of these addition problems step-by-step.
### a) [tex]\( x+3 \)[/tex] and [tex]\( x+2 \)[/tex]
To add these expressions, combine like terms (those involving the same variable):
[tex]\[ (x + 3) + (x + 2) = x + x + 3 + 2 \][/tex]
[tex]\[ = 2x + 5 \][/tex]
### b) [tex]\( 2x + 5 \)[/tex] and [tex]\( 3x \)[/tex]
To add these expressions, combine the terms with [tex]\( x \)[/tex] and the constants separately:
[tex]\[ (2x + 5) + (3x) = 2x + 3x + 5 \][/tex]
[tex]\[ = 5x + 5 \][/tex]
### d) [tex]\( 4 + 3a \)[/tex] and [tex]\( 5 - 8a \)[/tex]
To add these expressions, combine the like terms involving [tex]\( a \)[/tex] and the constants:
[tex]\[ (4 + 3a) + (5 - 8a) = 4 + 5 + 3a - 8a \][/tex]
[tex]\[ = 9 - 5a \][/tex]
### e) [tex]\( 9m - 2 \)[/tex] and [tex]\( m \)[/tex]
To add these expressions, combine the like terms involving [tex]\( m \)[/tex] and the constants:
[tex]\[ (9m - 2) + m = 9m + m - 2 \][/tex]
[tex]\[ = 10m - 2 \][/tex]
### 8) [tex]\( 5a + 4b \)[/tex] and [tex]\( 7a + 2b \)[/tex]
To add these expressions, combine the terms involving [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ (5a + 4b) + (7a + 2b) = 5a + 7a + 4b + 2b \][/tex]
[tex]\[ = 12a + 6b \][/tex]
### h) [tex]\( 8a - 5b \)[/tex] and [tex]\( a \)[/tex]
To add these expressions, combine the terms involving [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ (8a - 5b) + a = 8a + a - 5b \][/tex]
[tex]\[ = 9a - 5b \][/tex]
### (1) [tex]\( 3a^2 - 5b^2 \)[/tex] and [tex]\( 2a^2 + 7b^2 \)[/tex]
To add these expressions, combine the terms involving [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex]:
[tex]\[ (3a^2 - 5b^2) + (2a^2 + 7b^2) = 3a^2 + 2a^2 - 5b^2 + 7b^2 \][/tex]
[tex]\[ = 5a^2 + 2b^2 \][/tex]
### 11) [tex]\( x^8 + x - 2 \)[/tex] and [tex]\( 2x^2 - 5x + 7 \)[/tex]
To add these expressions, combine like terms, though in this case, there are no like terms for [tex]\( x^8 \)[/tex] and [tex]\( 2x^2 \)[/tex]. So just write them together:
[tex]\[ (x^8 + x - 2) + (2x^2 - 5x + 7) = x^8 + 2x^2 + x - 5x - 2 + 7 \][/tex]
[tex]\[ = x^8 + 2x^2 - 4x + 5 \][/tex]
### 7) [tex]\( 8abc - 5ab + 3a \)[/tex] and [tex]\( 2a + 7ab - 4abc \)[/tex]
To add these expressions, combine the like terms involving [tex]\( abc \)[/tex], [tex]\( ab \)[/tex], and [tex]\( a \)[/tex]:
[tex]\[ (8abc - 5ab + 3a) + (2a + 7ab - 4abc) = 8abc - 4abc - 5ab + 7ab + 3a + 2a \][/tex]
[tex]\[ = 4abc + 2ab + 5a \][/tex]
### 1) Subtracting Polynomials
For this last task, since the actual expressions to subtract were not provided in this part, ensure you understand the general method:
If you need to subtract [tex]\( P(x) \)[/tex] from [tex]\( Q(x) \)[/tex], you should change the signs of [tex]\( P(x) \)[/tex] and then add [tex]\( Q(x) \)[/tex] to [tex]\(-P(x) \)[/tex].
Example:
[tex]\[ P(x) = 4x + 5 \][/tex]
[tex]\[ Q(x) = x + 3 \][/tex]
[tex]\( Q(x) - P(x) \)[/tex]:
[tex]\[ (x + 3) - (4x + 5) \][/tex]
[tex]\[ = x + 3 - 4x - 5 \][/tex]
[tex]\[ = -3x - 2 \][/tex]
Make sure to follow a similar approach to solve subtraction problems!
