High School

We appreciate your visit to Combine the polynomials tex 5x 4 9x 3 8x 8x 4 9x 3 7 tex tex x 4 5x 3 2x 2 5x 4 5x. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Combine the polynomials.

[tex]
\[
(5x^4 + 9x^3 + 8x) + (8x^4 - 9x^3 + 7) =
\]
[/tex]

[tex]
\[
(-x^4 - 5x^3 + 2x^2) - (-5x^4 + 5x^3 - 4x) =
\]
[/tex]

Answer :

We are given two polynomial operations to simplify. Follow the steps below:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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]

Thanks for taking the time to read Combine the polynomials tex 5x 4 9x 3 8x 8x 4 9x 3 7 tex tex x 4 5x 3 2x 2 5x 4 5x. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada