ADDING POLYNOMIALS
How to add two polynomials?
Adding polynomials is nothing but combining the like terms.
Let us consider the following problem.
Add : (3x³-5x²+ 2x-7) and (4x²+x-8)
Here we give step by step explanation for adding the above two polynomials.
Step 1 :
Before going to add two polynomials, first we have to arrange the given polynomials one by one from highest power to lowest power.
(3x³-5x²+ 2x-7) and (4x²+x-8)
The two given polynomials are already in the arranged form.So we can leave it as it is.
Step 2 :
Now we have to write the like terms together starting from the highest power to lowest power.
= (3x³-5x²+ 2x-7) + (4x²+ x - 8)
Step 3:
Combine the like terms (Add or subtract) based on the signs of those terms.
In the second polynomial,we do not have x³ term,so we have to consider that there is zero x³.
So the final answer is 3x³ - 1x² + 3x - 15
To have better understanding, let us look at some more examples on "Adding polynomials"
Example 1 :
Add ( 7p³ + 4p²- 8p + 1 ) and (3p³- 5p²- 10p + 5)
Solution :
Step 1:
The two given polynomials are already in the arranged form.So we can leave it as it is.
= ( 7p³ + 4p²- 8p + 1) + (3p³ - 5p² - 10p + 5)
Step 2 :
Now we have to write the like terms together starting from the highest power to lowest power.
= 7p³ + 3p³ + 4p²- 5p²- 8p - 10p + 1 + 5
So the final answer is 10p³- 1p²- 18p + 6
Let us look at the next example on "Adding polynomials"
Example 2 :
Add ( 2x³ + 5x² - 2x + 7 ) and ( x³ + 4x² - x + 6)
Solution :
= ( 2x³ + 5x² - 2x + 7 ) + ( x³ + 4x² - x + 6)
= 2x³ + 5x² - 2x + 7 + x³ + 4x² - x + 6
= 2x³ + x³ + 5x² + 4x² - 2x - x + 7 + 6
= 3x³ + 9x² - 3x + 13
Let us look at the next example on "Adding polynomials"
Example 3 :
Add ( 3x³ - 2x² - x + 4 ) and ( 2x³ + 7x² - 3x - 3 )
Solution :
= (3x³ - 2x² - x + 4) + (2x³ + 7x² - 3x - 3)
= 3 x³ - 2 x² - x + 4 + 2 x³ + 7 x² - 3 x - 3
= 3x³ + 2x³ - 2x² + 7x² - x - 3x + 4 - 3
= 5x³ + 5x² - 4x + 1
Let us look at the next example on "Adding polynomials"
Example 4 :
Add 2( x³ - x² + 6x - 2 ) and ( 5x⁶ + 7x⁵ - 3x - 3 )
Solution :
= 2( x³ - x² + 6 x - 2 ) + ( 5 x⁶ + 7 x⁵ - 3 x - 3 )
= 2x³ - 2x² + 12x - 4 + 5x⁶ + 7x⁵ - 3x - 3
= 5x⁶ + 7x⁵ + 2x³ - 2x² + 12x - 3x - 4 - 3
= 5x⁶ + 7x⁵ + 2x³ - 2x² + 9x - 7
Let us look at the next example on "Adding polynomials"
Example 5 :
Add -1( x⁶ + x³ + 6x² - 2 ) and 2( 5x⁶ + 7x⁵ - 3x - 3 )
Solution :
= -1( x⁶ + x³ + 6x² - 2 ) + 2( 5x⁶ + 7x⁵ - 3x - 3 )
= -x⁶ - x³ - 6x² + 2 + 10x⁶ + 14x⁵ - 6x - 6
= -x⁶ + 10x⁶ + 14x⁵ - x³ - 6x² - 6x + 2 - 6
= 9x⁶ + 14x⁵ - x³ - 6x² - 6x - 4
Let us look at the next example on "Adding polynomials"
Example 6 :
Add 5( 5x⁶ + 2x³ - 6x² - 2 ) + 6(-3x⁶ + 2x⁵ + 2x + 1 )
Solution :
= 5( 5x⁶ + 2x³ - 6x² - 2 ) + 6( -3x⁶ + 2x⁵ + 2x + 1 )
= 25x⁶ + 10x³ - 30x² - 10 -18x⁶ + 12x⁵ + 12x + 6
= 25x⁶ -18x⁶ + 12x⁵ + 10x³ - 30x² + 12x -10 + 6
= 7x⁶ + 12x⁵ + 10x³ - 30x² + 12x - 4
Let us look at the next example on "Adding polynomials"
Example 7 :
Add -2 ( 2x⁴ - 2x³ - x² + 5 ) and 3 ( 2x⁴ - 2x² - 3 )
Solution :
= -2( 2x⁴ - 2x³ - x² + 5 ) + 3( 2x⁴ - 2x² - 3 )
= -4x⁴ + 4x³ + 2x² -10 + 6x⁴ - 6x² - 9
= -4x⁴ + 6x⁴ + 4x³ + 2x² - 6x² -10 - 9
= 2x⁴ + 4x³ - 4x² - 19
Let us look at the next example on "Adding polynomials"
Example 8 :
Add 5( x⁴ - x³ + 5 ) and 2( x⁴ - 5x² - 7 )
Solution :
= 5( x⁴ - x³ + 5 ) + 2( x⁴ - 5x² - 7 )
= 5x⁴ - 5 x³ + 25 + 2x⁴ - 10x² - 14
= 5x⁴ + 2x⁴ - 5x³ - 10x² + 25 - 14
= 7x⁴ - 15x³ + 11
Let us look at the next example on "Adding polynomials"
Example 9 :
Add 3( 6x⁴ - 2x³ - 3 ) and 2( 2x⁴ - x² - 8 )
Solution :
= 3 ( 6x⁴ - 2x³ - 3 ) + 2 ( 2x⁴ - x² - 8 )
= 18x⁴ - 6x³ - 9 + 4x⁴ -2x² - 16
= 18x⁴ + 4x⁴ - 6x³ -2x² - 9 - 16
= 22 x⁴ - 6x³ -2x² - 25
Let us look at the next example on "Adding polynomials"
Example 10 :
Add (6x⁷-2x⁶-3x³+2x²) and 2(2x⁴+5x⁷+ 3x⁶+ x³+x²)
Solution :
= ( 6x⁷- 2x⁶- 3x³+ 2x²) + 2( 2x⁴ + 5x⁷ + 3x⁶ + x³ + x² )
= 6x⁷ - 2x⁶ - 3x³ + 2x² + 4x⁴ + 10x⁷ + 6x⁶ + 2x³ + 2x²
= 6x⁷ + 10x⁷- 2x⁶ + 6x⁶ + 4x⁴ - 3x³ + 2x³ + 2x² + 2x²
= 16x⁷ + 4x⁶ + 4x⁴ - x³ + 4x²
Let us look at the next example on "Adding polynomials"
Example 11 :
Add (x⁷-3x⁶-2x³+x²) and 5 (3x⁴+15x⁷+4x⁶+2x³+6x²)
Solution :
= (x⁷-3x⁶-2x³+x²) + 5 (3x⁴ + 15x⁷ + 4x⁶ + 2x³+ 6x² )
= x⁷- 3x⁶ - 2x³ + x² + 15x⁴ + 75x⁷ + 20x⁶ + 10x³ + 30x²
= x⁷ + 75x⁷- 3x⁶ + 20x⁶ - 2x³ + 10x³ + x² + 30x²
= 76x⁷ + 17x⁶ + 8x³ + 31x²
After having gone through the examples explained above, we hope that the students would have understood the stuff "Adding polynomials"
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