Math / Algebra

QuestionAdditionne.
23. $\left(m^{2}+3 m-5\right)+\left(2 m^{2}-5 m+7\right)$
24. $\left(5 a+3 a^{2}-2\right)+\left(a^{2}+3 a+4\right)$
25. $\left(b+5-2 b^{2}\right)+\left(1-2 b+b^{2}\right)$

Studdy Solution

STEP 1

What is this asking? We're asked to add some polynomials together, combining like terms! Watch out! Be careful with the signs when combining terms, especially when subtracting!

STEP 2

1. Problem 23
2. Problem 24
3. Problem 25

STEP 3

Alright, let's start with problem 23!
We have $(m^2 + 3m - 5) + (2m^2 - 5m + 7)$ .

STEP 4

First, let's remove those parentheses since we're adding: $m^2 + 3m - 5 + 2m^2 - 5m + 7$ .

STEP 5

Now, let's group our like terms together.
Remember, "like terms" are terms with the same variable raised to the same power.
We've got $m^2$ terms, $m$ terms, and constant terms: $m^2 + 2m^2 + 3m - 5m - 5 + 7$ .
See how we kept the signs with the terms?
Super important!

STEP 6

Time to combine those like terms! $m^2 + 2m^2$ gives us $3m^2$ . $3m - 5m$ gives us $-2m$ .
And $-5 + 7$ gives us $2$ .

STEP 7

Putting it all together, our final answer is $3m^2 - 2m + 2$ !
Boom!

STEP 8

On to problem 24!
We have $(5a + 3a^2 - 2) + (a^2 + 3a + 4)$ .

STEP 9

Let's get rid of those parentheses: $5a + 3a^2 - 2 + a^2 + 3a + 4$ .

STEP 10

Group the like terms: $3a^2 + a^2 + 5a + 3a - 2 + 4$ .

STEP 11

Combine like terms: $3a^2 + a^2$ is $4a^2$ , $5a + 3a$ is $8a$ , and $-2 + 4$ is $2$ .

STEP 12

So, our simplified expression is $4a^2 + 8a + 2$ !
Fantastic!

STEP 13

Last one, problem 25!
We've got $(b + 5 - 2b^2) + (1 - 2b + b^2)$ .

STEP 14

Drop those parentheses: $b + 5 - 2b^2 + 1 - 2b + b^2$ .

STEP 15

Rearrange and group like terms: $-2b^2 + b^2 + b - 2b + 5 + 1$ .

STEP 16

Combine the like terms: $-2b^2 + b^2$ gives us $-b^2$ , $b - 2b$ gives us $-b$ , and $5 + 1$ gives us $6$ .

STEP 17

Our final answer is $-b^2 - b + 6$ !
We did it!

STEP 18

23. $3m^2 - 2m + 2$
24. $4a^2 + 8a + 2$
25. $-b^2 - b + 6$

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Studdy Solution

STEP 1

What is this asking? We're asked to add some polynomials together, combining like terms! Watch out! Be careful with the signs when combining terms, especially when subtracting!

STEP 2

1. Problem 232. Problem 243. Problem 25

STEP 3

Alright, let's **start** with problem 23!We have (m2+3m−5)+(2m2−5m+7)(m^2 + 3m - 5) + (2m^2 - 5m + 7)(m2+3m−5)+(2m2−5m+7).

STEP 4

First, let's **remove** those parentheses since we're adding: m2+3m−5+2m2−5m+7m^2 + 3m - 5 + 2m^2 - 5m + 7m2+3m−5+2m2−5m+7.

STEP 5

STEP 6

Time to **combine** those like terms! m2+2m2m^2 + 2m^2m2+2m2 gives us 3m23m^23m2. 3m−5m3m - 5m3m−5m gives us −2m-2m−2m.And −5+7-5 + 7−5+7 gives us 222.

STEP 7

Putting it all together, our **final answer** is 3m2−2m+23m^2 - 2m + 23m2−2m+2!Boom!

STEP 8

On to problem 24!We have (5a+3a2−2)+(a2+3a+4)(5a + 3a^2 - 2) + (a^2 + 3a + 4)(5a+3a2−2)+(a2+3a+4).

STEP 9

Let's **get rid** of those parentheses: 5a+3a2−2+a2+3a+45a + 3a^2 - 2 + a^2 + 3a + 45a+3a2−2+a2+3a+4.

STEP 10

**Group** the like terms: 3a2+a2+5a+3a−2+43a^2 + a^2 + 5a + 3a - 2 + 43a2+a2+5a+3a−2+4.

STEP 11

**Combine** like terms: 3a2+a23a^2 + a^23a2+a2 is 4a24a^24a2, 5a+3a5a + 3a5a+3a is 8a8a8a, and −2+4-2 + 4−2+4 is 222.

STEP 12

So, our **simplified expression** is 4a2+8a+24a^2 + 8a + 24a2+8a+2!Fantastic!

STEP 13

Last one, problem 25!We've got (b+5−2b2)+(1−2b+b2)(b + 5 - 2b^2) + (1 - 2b + b^2)(b+5−2b2)+(1−2b+b2).

STEP 14

**Drop** those parentheses: b+5−2b2+1−2b+b2b + 5 - 2b^2 + 1 - 2b + b^2b+5−2b2+1−2b+b2.

STEP 15

**Rearrange** and group like terms: −2b2+b2+b−2b+5+1-2b^2 + b^2 + b - 2b + 5 + 1−2b2+b2+b−2b+5+1.

STEP 16

**Combine** the like terms: −2b2+b2-2b^2 + b^2−2b2+b2 gives us −b2-b^2−b2, b−2bb - 2bb−2b gives us −b-b−b, and 5+15 + 15+1 gives us 666.

STEP 17

Our **final answer** is −b2−b+6-b^2 - b + 6−b2−b+6!We did it!

STEP 18

23. 3m2−2m+23m^2 - 2m + 23m2−2m+224. 4a2+8a+24a^2 + 8a + 24a2+8a+225. −b2−b+6-b^2 - b + 6−b2−b+6

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1. Problem 23
2. Problem 24
3. Problem 25

Alright, let's start with problem 23!
We have $(m^2 + 3m - 5) + (2m^2 - 5m + 7)$ .

First, let's remove those parentheses since we're adding: $m^2 + 3m - 5 + 2m^2 - 5m + 7$ .

Time to combine those like terms! $m^2 + 2m^2$ gives us $3m^2$ . $3m - 5m$ gives us $-2m$ .
And $-5 + 7$ gives us $2$ .

Putting it all together, our final answer is $3m^2 - 2m + 2$ !
Boom!

On to problem 24!
We have $(5a + 3a^2 - 2) + (a^2 + 3a + 4)$ .

Let's get rid of those parentheses: $5a + 3a^2 - 2 + a^2 + 3a + 4$ .

Group the like terms: $3a^2 + a^2 + 5a + 3a - 2 + 4$ .

Combine like terms: $3a^2 + a^2$ is $4a^2$ , $5a + 3a$ is $8a$ , and $-2 + 4$ is $2$ .

So, our simplified expression is $4a^2 + 8a + 2$ !
Fantastic!

Last one, problem 25!
We've got $(b + 5 - 2b^2) + (1 - 2b + b^2)$ .

Drop those parentheses: $b + 5 - 2b^2 + 1 - 2b + b^2$ .

Rearrange and group like terms: $-2b^2 + b^2 + b - 2b + 5 + 1$ .

Combine the like terms: $-2b^2 + b^2$ gives us $-b^2$ , $b - 2b$ gives us $-b$ , and $5 + 1$ gives us $6$ .

Our final answer is $-b^2 - b + 6$ !
We did it!

23. $3m^2 - 2m + 2$
24. $4a^2 + 8a + 2$
25. $-b^2 - b + 6$