To know adding and subtracting polynomials, we have to know about two terms.
i) Like terms
ii) Unlike terms
What are like terms ?
Like terms are the terms which have the same variables with same exponent for each variable.
What are unlike terms ?
Unlike terms will not have same variables or they will not have same exponents.
Simplify the following.
Problem 1 :
(5x^{2} – 2x + 7) + (x^{3} – 5x^{2} – x + 3)
Solution :
Given, (5x^{2} – 2x + 7) + (x^{3} – 5x^{2} – x + 3)
= 5x^{2} – 2x + 7 + x^{3} – 5x^{2} – x + 3
By combining like terms,
= x^{3} - 2x – x + 7 + 3
= x^{3} – 3x + 10
Problem 2 :
(x^{3} – x^{2} + x + 1) + (x^{3} + x^{2} - x - 1)
Solution :
Given, (x^{3} – x^{2} + x + 1) + (x^{3} + x^{2} - x - 1)
= x^{3} – x^{2} + x + 1 + x^{3} + x^{2} – x – 1
By combining like terms,
= x^{3} + x^{3}
= 2x^{3}
Problem 3 :
(2x^{4} + x^{2} - 1) + (x^{4} – x^{3} + x + 1)
Solution :
Given, (2x^{4} + x^{2} - 1) + (x^{4} – x^{3} + x + 1)
= 2x^{4} + x^{2} - 1 + x^{4} – x^{3} + x + 1
By combining like terms,
= 2x^{4} + x^{4} – x^{3} + x^{2} + x
= 3x^{4} – x^{3} + x^{2} + x
Problem 4 :
(x^{4} - 2x^{3} + 3x^{2} – 4x + 5) + (5x^{3} – x^{2} + 7x - 3)
Solution :
Given, (x^{4} - 2x^{3} + 3x^{2} – 4x + 5) + (5x^{3} – x^{2} + 7x - 3)
= x^{4} - 2x^{3} + 3x^{2} – 4x + 5 + 5x^{3} – x^{2} + 7x - 3
By combining like terms,
= x^{4} – 2x^{3} + 5x^{3} + 3x^{2} – x^{2} – 4x + 7x + 5 – 3
= x^{4} + 3x^{3} + 2x^{2} + 3x + 2
Problem 5 :
(2x^{3} + 3x^{2} – 5x + 1) + (6 – 2x + 3x^{2} – x^{3})
Solution :
Given, (2x^{3} + 3x^{2} – 5x + 1) + (6 – 2x + 3x^{2} – x^{3})
= 2x^{3} + 3x^{2} – 5x + 1 + 6 – 2x + 3x^{2} – x^{3}
By combining like terms,
= 2x^{3} – x^{3} + 3x^{2} + 3x^{2} – 5x – 2x + 1 + 6
= x^{3} + 6x^{2} – 7x + 7
Problem 6 :
(4x^{2} – x + 3) – (3x^{2} + x + 1)
Solution :
Given, (4x^{2} – x + 3) – (3x^{2} + x + 1)
= 4x^{2} – x + 3 – 3x^{2} - x - 1
By combining like terms,
= 4x^{2} – 3x^{2} – x – x + 3 – 1
= x^{2} – 2x + 2
Problem 7 :
(2x^{3} + x^{2} + 5x - 7) – (x^{3} – 2x^{2} + 5x + 4)
Solution :
Given, (2x^{3} + x^{2} + 5x - 7) – (x^{3} – 2x^{2} + 5x + 4)
= 2x^{3} + x^{2} + 5x - 7 – x^{3} + 2x^{2} - 5x – 4
By combining like terms,
= 2x^{3} - x^{3} + x^{2} + 2x^{2} – 7 – 4
= x^{3} + 3x^{2} – 11
Problem 8 :
(9x^{4} + x^{3} + 2x - 3) – (5x^{4} + 7x^{2} – 2x + 3)
Solution :
Given, (9x^{4} + x^{3} + 2x - 3) – (5x^{4} + 7x^{2} – 2x + 3)
= 9x^{4} + x^{3} + 2x - 3 – 5x^{4} - 7x^{2} + 2x - 3
By combining like terms,
= 9x^{4} – 5x^{4} + x^{3} – 7x^{2} + 2x + 2x – 3 – 3
= 4x^{4} + x^{3} – 7x^{2} + 4x – 6
Problem 9 :
(5x^{2} + 7x + 1) + (2x^{2} + x - 3) + (x^{2} – 10x + 7)
Solution :
Given, (5x^{2} + 7x + 1) + (2x^{2} + x - 3) + (x^{2} – 10x + 7)
= 5x^{2} + 7x + 1 + 2x^{2} + x - 3 + x^{2} – 10x + 7
By combining like terms,
= 5x^{2} + 2x^{2} + x^{2} + 7x + x – 10x + 1 + 7
= 8x^{2} - 2x + 8
Problem 10 :
(x^{3} – x + 3) + (x^{2} – 3x + 4) + (2x^{3} – x^{2} + 5)
Solution :
Given, (x^{3} – x + 3) + (x^{2} – 3x + 4) + (2x^{3} – x^{2} + 5)
= x^{3} – x + 3 + x^{2} – 3x + 4 + 2x^{3} – x^{2} + 5
By combining like terms,
= x^{3} + 2x^{3} – x – 3x + 3 + 4 + 5
= 3x^{3} – 4x + 11
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May 21, 24 08:51 AM
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