Answer:
1) 5y + 5z
2) 10x⁴ - 10x³ + 2x² + 18x - 11
Step-by-step explanation:
Given the subtraction of the following polynomial expressions:
(1) -3x - 4y + 11z from -9y + 6z - 3x
In order to make it easier for us to perform the required mathematical operations, we must first rearrange the terms in the subtrahend by alphabetical order.
-3x - 4y + 11z
-3x - 9y + 6z ⇒ This is the subtrahend.
Now, we can finally perform the subtraction on both trinomials:
[tex]\displaystyle\mathsf{\left \ \quad\:\:\:\:{-3x - 4y + 11z} \atop -\quad{\underline{-3x - 9y + 6z\:\:\underline}} \right.}[/tex]
In the subtrahend, the coefficients of x and y are both negative. Thus, performing the subtraction operations on these coefficients transforms their sign into positive.
[tex]\displaystyle\mathsf{\left \ \quad\:\:\:\:{-3x - 4y + 11z} \atop -\quad{\underline{-3x - 9y + 6z\:\:\underline}}\right.} \\\qquad\sf {\qquad\:\:\:0x\:+\:5y\:+5z[/tex]
The difference is: 5y + 5z.
(2) 3x⁴- 4x³ + 7x - 2 from 9 - 7x⁴ + 6x³- 2x² - 11x
Similar to the how we arranged the given trinomials in Question 1, we must rearrange the given polynomials in descending degree of terms before subtracting like terms.
3x⁴- 4x³ + 7x - 2 ⇒ Already in descending order (degree).
9 - 7x⁴ + 6x³- 2x² - 11x ⇒ -7x⁴ + 6x³- 2x² - 11x + 9
In subtracting polynomials, we can only subtract like terms, which are terms that have the same variables and exponents.
[tex]\displaystyle\mathsf{\left \ \quad\:\:{3x^4\:-4x^3\:+\:0x^2\:+\:7x\:-\:2} \atop -\quad{\underline{-7x^4\:+6x^3\:-2x^2\:-11x\:+\:9 \:\:\underline}}\right.}[/tex]
In the minuend, I added the "0x²" to make it less-confusing for us to perform the subtraction operations.
The same rules apply in terms of coefficients with negative signs in the subtrahend, such as: -7x⁴, - 2x², and - 11x ⇒ their coefficients turn into positive when performing subtraction.
[tex]\displaystyle\mathsf{\left \ \quad\:\:{3x^4\:-4x^3\:+\:0x^2\:+\:7x\:-\:2} \atop -\quad{\underline{-7x^4\:+6x^3\:-2x^2\:-11x\:+\:9 \:\:\underline}}\right.} \\\qquad\sf {\qquad\:\:10x^4-10x^3+2x^2+18x\:-11[/tex]
Therefore, the difference is: 10x⁴ - 10x³ + 2x² + 18x - 11.
