Answer :
Sure! Let's solve the problem step-by-step:
We are given two polynomials:
1. The polynomial to start with:
[tex]\(5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4\)[/tex]
2. The difference we want to obtain:
[tex]\(19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4\)[/tex]
We need to find a polynomial that, when subtracted from the first polynomial, results in the second polynomial. Mathematically, we can write this as:
[tex]\[
\text{First Polynomial} - \text{Unknown Polynomial} = \text{Difference Polynomial}
\][/tex]
Rearranging this equation to solve for the unknown polynomial, we get:
[tex]\[
\text{Unknown Polynomial} = \text{First Polynomial} - \text{Difference Polynomial}
\][/tex]
Let's write down these polynomials in standard form to make it easier to subtract them:
First Polynomial:
[tex]\[
5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4
\][/tex]
Difference Polynomial:
[tex]\[
19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4
\][/tex]
Now we subtract the Difference Polynomial from the First Polynomial term by term:
1. [tex]\( - 12x^5 - (-19x^5) = -12x^5 + 19x^5 = 7x^5 \)[/tex]
2. [tex]\( - 6x^4 - (-8x^4) = -6x^4 + 8x^4 = 2x^4 \)[/tex]
3. [tex]\( 5x^3 - 8x^3 = -3x^3 \)[/tex]
4. [tex]\( -x^2 - (-2x^2) = -x^2 + 2x^2 = x^2 \)[/tex]
5. [tex]\( -12x - (-18x) = -12x + 18x = 6x \)[/tex]
6. [tex]\( 9 - 19 = -10 \)[/tex]
Putting all these together, we get the polynomial that should be subtracted:
[tex]\[
7x^5 + 2x^4 - 3x^3 + x^2 + 6x - 10
\][/tex]
So, the polynomial that you need to subtract from [tex]\(5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4\)[/tex] to get the difference of [tex]\(19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4\)[/tex] is:
[tex]\[
7x^5 + 2x^4 - 3x^3 + x^2 + 6x - 10
\][/tex]
We are given two polynomials:
1. The polynomial to start with:
[tex]\(5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4\)[/tex]
2. The difference we want to obtain:
[tex]\(19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4\)[/tex]
We need to find a polynomial that, when subtracted from the first polynomial, results in the second polynomial. Mathematically, we can write this as:
[tex]\[
\text{First Polynomial} - \text{Unknown Polynomial} = \text{Difference Polynomial}
\][/tex]
Rearranging this equation to solve for the unknown polynomial, we get:
[tex]\[
\text{Unknown Polynomial} = \text{First Polynomial} - \text{Difference Polynomial}
\][/tex]
Let's write down these polynomials in standard form to make it easier to subtract them:
First Polynomial:
[tex]\[
5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4
\][/tex]
Difference Polynomial:
[tex]\[
19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4
\][/tex]
Now we subtract the Difference Polynomial from the First Polynomial term by term:
1. [tex]\( - 12x^5 - (-19x^5) = -12x^5 + 19x^5 = 7x^5 \)[/tex]
2. [tex]\( - 6x^4 - (-8x^4) = -6x^4 + 8x^4 = 2x^4 \)[/tex]
3. [tex]\( 5x^3 - 8x^3 = -3x^3 \)[/tex]
4. [tex]\( -x^2 - (-2x^2) = -x^2 + 2x^2 = x^2 \)[/tex]
5. [tex]\( -12x - (-18x) = -12x + 18x = 6x \)[/tex]
6. [tex]\( 9 - 19 = -10 \)[/tex]
Putting all these together, we get the polynomial that should be subtracted:
[tex]\[
7x^5 + 2x^4 - 3x^3 + x^2 + 6x - 10
\][/tex]
So, the polynomial that you need to subtract from [tex]\(5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4\)[/tex] to get the difference of [tex]\(19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4\)[/tex] is:
[tex]\[
7x^5 + 2x^4 - 3x^3 + x^2 + 6x - 10
\][/tex]
