Answer :
We are given two polynomial operations to simplify. Follow the steps below:
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Step 1. Combine the first set of polynomials:
[tex]$$
\left(5x^4 + 9x^3 + 8x\right) + \left(8x^4 - 9x^3 + 7\right)
$$[/tex]
1. Combine the [tex]$x^4$[/tex] terms:
[tex]$$5x^4 + 8x^4 = 13x^4.$$[/tex]
2. Combine the [tex]$x^3$[/tex] terms:
[tex]$$9x^3 + (-9x^3) = 0x^3.$$[/tex]
3. Combine the [tex]$x$[/tex] terms (note, the second polynomial has no [tex]$x$[/tex] term):
[tex]$$8x + 0 = 8x.$$[/tex]
4. Combine the constant terms:
[tex]$$0 + 7 = 7.$$[/tex]
Thus, the combined expression is:
[tex]$$
13x^4 + 8x + 7.
$$[/tex]
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Step 2. Combine the second set of polynomials:
[tex]$$
\left(-x^4 - 5x^3 + 2x^2\right) - \left(-5x^4 + 5x^3 - 4x\right)
$$[/tex]
Note that subtracting a polynomial means changing the sign of each term in that polynomial. Rewrite the expression by distributing the negative sign:
[tex]$$
-x^4 - 5x^3 + 2x^2 \;+\; 5x^4 - 5x^3 + 4x.
$$[/tex]
Now, combine like terms:
1. Combine the [tex]$x^4$[/tex] terms:
[tex]$$-x^4 + 5x^4 = 4x^4.$$[/tex]
2. Combine the [tex]$x^3$[/tex] terms:
[tex]$$-5x^3 - 5x^3 = -10x^3.$$[/tex]
3. Combine the [tex]$x^2$[/tex] terms:
[tex]$$2x^2 \quad (\text{no corresponding term to combine}).$$[/tex]
4. Combine the [tex]$x$[/tex] terms:
[tex]$$0 + 4x = 4x.$$[/tex]
Since there is no constant term, the final result is:
[tex]$$
4x^4 - 10x^3 + 2x^2 + 4x.
$$[/tex]
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Final Answers:
1. For the first set, the simplified polynomial is
[tex]$$
13x^4 + 8x + 7.
$$[/tex]
2. For the second set, the simplified polynomial is
[tex]$$
4x^4 - 10x^3 + 2x^2 + 4x.
$$[/tex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Step 1. Combine the first set of polynomials:
[tex]$$
\left(5x^4 + 9x^3 + 8x\right) + \left(8x^4 - 9x^3 + 7\right)
$$[/tex]
1. Combine the [tex]$x^4$[/tex] terms:
[tex]$$5x^4 + 8x^4 = 13x^4.$$[/tex]
2. Combine the [tex]$x^3$[/tex] terms:
[tex]$$9x^3 + (-9x^3) = 0x^3.$$[/tex]
3. Combine the [tex]$x$[/tex] terms (note, the second polynomial has no [tex]$x$[/tex] term):
[tex]$$8x + 0 = 8x.$$[/tex]
4. Combine the constant terms:
[tex]$$0 + 7 = 7.$$[/tex]
Thus, the combined expression is:
[tex]$$
13x^4 + 8x + 7.
$$[/tex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Step 2. Combine the second set of polynomials:
[tex]$$
\left(-x^4 - 5x^3 + 2x^2\right) - \left(-5x^4 + 5x^3 - 4x\right)
$$[/tex]
Note that subtracting a polynomial means changing the sign of each term in that polynomial. Rewrite the expression by distributing the negative sign:
[tex]$$
-x^4 - 5x^3 + 2x^2 \;+\; 5x^4 - 5x^3 + 4x.
$$[/tex]
Now, combine like terms:
1. Combine the [tex]$x^4$[/tex] terms:
[tex]$$-x^4 + 5x^4 = 4x^4.$$[/tex]
2. Combine the [tex]$x^3$[/tex] terms:
[tex]$$-5x^3 - 5x^3 = -10x^3.$$[/tex]
3. Combine the [tex]$x^2$[/tex] terms:
[tex]$$2x^2 \quad (\text{no corresponding term to combine}).$$[/tex]
4. Combine the [tex]$x$[/tex] terms:
[tex]$$0 + 4x = 4x.$$[/tex]
Since there is no constant term, the final result is:
[tex]$$
4x^4 - 10x^3 + 2x^2 + 4x.
$$[/tex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Final Answers:
1. For the first set, the simplified polynomial is
[tex]$$
13x^4 + 8x + 7.
$$[/tex]
2. For the second set, the simplified polynomial is
[tex]$$
4x^4 - 10x^3 + 2x^2 + 4x.
$$[/tex]
