Answer :
Let's solve the given problems step-by-step:
1) Add the two polynomials:
The expression we need to simplify is:
[tex]\[
(6x^3 - 8x^2 + 4) + (7 + 2x^2 - 7x^4)
\][/tex]
To simplify, combine like terms:
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-7x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(6x^3\)[/tex] (no other [tex]\(x^3\)[/tex] terms to combine)
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-8x^2 + 2x^2 = -6x^2\)[/tex]
- Combine the constant terms: [tex]\(4 + 7 = 11\)[/tex]
Putting it all together, the simplified expression is:
[tex]\[
-7x^4 + 6x^3 - 6x^2 + 11
\][/tex]
So the correct answer is option B) [tex]\(-7x^4 + 6x^3 - 6x^2 + 11\)[/tex].
2) Expand the expression [tex]\( (3n - 7)(8n - 4) \)[/tex]:
Apply the distributive property (also known as the FOIL method for binomials):
- First: [tex]\((3n) \cdot (8n) = 24n^2\)[/tex]
- Outer: [tex]\((3n) \cdot (-4) = -12n\)[/tex]
- Inner: [tex]\((-7) \cdot (8n) = -56n\)[/tex]
- Last: [tex]\((-7) \cdot (-4) = 28\)[/tex]
Now combine the like terms:
- Combine the [tex]\(n\)[/tex] terms: [tex]\(-12n - 56n = -68n\)[/tex]
So the expanded expression is:
[tex]\[
24n^2 - 68n + 28
\][/tex]
There you have the step-by-step solutions for both questions.
1) Add the two polynomials:
The expression we need to simplify is:
[tex]\[
(6x^3 - 8x^2 + 4) + (7 + 2x^2 - 7x^4)
\][/tex]
To simplify, combine like terms:
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-7x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(6x^3\)[/tex] (no other [tex]\(x^3\)[/tex] terms to combine)
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-8x^2 + 2x^2 = -6x^2\)[/tex]
- Combine the constant terms: [tex]\(4 + 7 = 11\)[/tex]
Putting it all together, the simplified expression is:
[tex]\[
-7x^4 + 6x^3 - 6x^2 + 11
\][/tex]
So the correct answer is option B) [tex]\(-7x^4 + 6x^3 - 6x^2 + 11\)[/tex].
2) Expand the expression [tex]\( (3n - 7)(8n - 4) \)[/tex]:
Apply the distributive property (also known as the FOIL method for binomials):
- First: [tex]\((3n) \cdot (8n) = 24n^2\)[/tex]
- Outer: [tex]\((3n) \cdot (-4) = -12n\)[/tex]
- Inner: [tex]\((-7) \cdot (8n) = -56n\)[/tex]
- Last: [tex]\((-7) \cdot (-4) = 28\)[/tex]
Now combine the like terms:
- Combine the [tex]\(n\)[/tex] terms: [tex]\(-12n - 56n = -68n\)[/tex]
So the expanded expression is:
[tex]\[
24n^2 - 68n + 28
\][/tex]
There you have the step-by-step solutions for both questions.
