9 Mathematics: Polynomial

1. Find the sum and the degree of the sum:
 p(x) =x3-5x2+x+2 and g(x) =x3-3x2+2x+1
Solution: 
  Given that, 
             p(x)=x3-5x2+x+2
      and g(x)=x3-3x2+2x+1.
Now, the sum of the polynomials is given by,
    p(x)+g(x)=x3-5x2+x+2+(x3-3x2+2x+1)
                        = 2x3-8x2+3x+3.
Here, the highest power of x in p(x)+g(x) = 3.
Hence, the degree of the sum is 3.

2. Find the sum and the degree of the sum:
 p(y)=y6-3y4 and g(y)= y4+y3+2y2-6
Solution: Given that,
           p(y)=y6-3y4
   and  g(y)= y4+y3+2y2-6.
Now, the sum of the polynomials is given by,
p(y)+g(y)=(y6-3y4)+(y4+y3+2y2-6)
                   = y6-2y4+y3+2y2-6.
Here, the highest power of y in p(y)+g(y)= 6.
Hence, the degree of the sum is 6.
 
3. Find the sum and the degree of the sum: 
p(x)=3x2+5x-4 and g(x)=6x4-x+2

4. Find the sum and the degree of the sum:
p(y)=3y5-4y3+y2+y-7 and g(y)=-3y5+4y3+6y2+7y+2

5. Find the sum and the degree of the sum: 
p(x)=3x2+5x-2 and g(x)=-3x2-5x+6

6. Find the sum and the degree of the sum:  p(t)=t2+t-7 and g(t)=t3+t2+3t+4

7. Find the difference and the degree of the difference: p(x)=x3-3x2+6 and g(x)= x2-x+4
Solution:
Given that,
p(x)=x3-3x2+6 and g(x)= x2-x+4 
Now, difference of p(x) and g(x) is given by,
p(x)-g(x) =(x3-3x2+6 )-(x2-x+4) 
                   = x3-3x2+6-x2+x-4
                   = x3-4x2+x+2.
Here, the highest power of x in p(x)-g(x)= 3.
Hence, the degree of the difference is 3.

8. Find the difference and the degree of the difference: 
p(t)= t4-3t3+2t+6 and g(t)= t4-3t3-6t+2

9. Subtract the second polynomial from the first and find the degree of this difference:
g(y) = y3+3y2+3y+1 and f(y)= y3-3y2+3y-1

10. If p(x) = x² – 3x + 2 and q(x) = x³ - 6x² + x + 1, find the product of p(x) and q(x) and degree of product.
 Solution:
Given p(x) = x² - 3x + 2
                q(x) = x³ - 6x² +  x + 1 
Now, 
Product p(x) and q(x) is
 p(x).q(x) = (x² - 3x + 2) (x³ - 6x² + x + 1 )
                   = x² (x³ - 6x² + x +1) - 3x (x³ - 6x² + x + 1) + 2(x³ - 6x² + x + 1)
                  = x⁵ - 6x⁴ + x³ + x² - 3x⁴ + 18x³ - 3x² - 3x +2x³- 12x² + 2x + 2 
                  = x⁵ - 10x⁴ - 21x³ - 14 x² – x + 2 .
Now degree of polynomial is 5. 

11. Polynomials h(x) = x² + (a + b) x² – 4x + 2 and k(x) = x³ + 5x² – (2a – b)x + 2 are equal to each other. Find the values of a and b.
 Solution:
Given h(x) = x³ + (a + b) x² - 4x + 2 and k(x) = x³ + 5x² - (2a - b)x + 2 are equal. 
So we can compare the corresponding coefficient. 
i.e  a + b = 5 ............... (i) 
and - (2a - b) = -4
    i.e. 2a - b =  4............... (ii) 
Adding   (i) &  (ii) 
  a + b = 5
+2a - b = 4
      3a = 9
       a = 3
Put,  a = 3 in  (i) 
     3 + b = 5
i.e. b= 5 – 3 = 2.
 ∴ a = 3 & b = 2 is the required.
 
12. Find the value of 'k'. If p(x) = x5 + x3 + (3k – 7)x² – 2x + 6 and q(x) = x⁵ + x³ - (1 - k) x² - 2x+ 6 are equal.
Solution: Given that, 
p(x) = x⁵ + x³ + (3k - 7) x² - 2x + 6 and
q(x) = x5 + x³ - (1 - k) x² - 2x + 6  are equal. 
So, we can compare the corresponding coefficient.
      3k - 7 = - (1 - k)
⇒3k - 7 = - 1+ k
⇒ 3k - k = -1+7
⇒ 2k = 6 
 ∴   k = 3.
Hence the value of k is 3. 
 
13. If the sum of f(x) = ax² – bx + 6 and g(x) = 3x² - 2x – 5 is h(x) = 5x² - 5x + 1 then find the values of the constants a, b and c.
Solution:
Given,
               f(x) = ax² – bx + 6
               g(x) = 3x² - 2x - 5
               h (x)  = 5x² - 5x + 1.

According to the question
      f(x) + g(x)  = h (x)
⇒ ax² - bx + c + 3x² - 2x - 5 = 5x² - 5x + 1
⇒ (a + 3) x² - (b +  2)x + (c - 5) =5x² - 5x + 1
Since, they are equal, so we can compare the coefficient. 
i.e.  a + 3 = 5      →    a = 5 - 3 = 2 
       b + 2 = 5     →     b = 5 - 2 = 3
       c – 5 = 1      →     c = 1 + 5 = 6.
 ∴ a = 2, b = 3 & c= 6 are required solution. 

14. If the sum of f(x) = ax² + bx + 6 and g(x) = 6x²- 3^x + c is h(x) = 10x² + 2x + 15 then find the values of the constants a, b and c.
 
15. If the sum of f(x) = Cx³ - 4x² + bx + 7 and g(x) = 9x³ + 6x² - 7x – a is h(x) = 11x³ + 2x² – x + 4 then find the values of the constants a, b and c.

16. If the sum of f(x) = ax² - 3x – 5 + 8x³ and g(x) = 5x² + bx +10 is h(x) = cx³ + 11x² – x + 5 then find the values of the constants a, b and c.

17. If p (x) = ax³ - bx² + cx – 6 is subtracted from q (x) = 5x³ - 2x² + 8x + 10 then the result is h(x) = 2x³ + 4x² + 4x + 16. Find the values of the constants a, b and c.
Solution: 
Given, p (x) = ax³ - bx² + cx – 6,
            q (x) = 5x³ - 2x² + 8x + 10 &
             h(x) = 2x³ + 4x² + 4x + 16.
According to question, 
      q (x) - p (x) = h(x)      
⇒   5x3 - 2x²  + 8x + 10 - (ax³ - bx² + cx - 6) = 2x³ + 4x²  + 4x + 16
⇒ (5 - a)x³ + (b - 2) x² +(8 - c)x + 10 + 6 = 2x³ + 4x²  + 4x + 16
Comparing the corresponding coefficient 
5 - a = 2         →   a = 5 - 2 = 3 
b - 2 = 4         →   b= 4 + 2 = 6 
8 - c = 4         →    c = 8 - 4 = 4 .
 ∴a = 3 , b = 6 & c = 4 are the required solution.

18. If the difference of p(x) = 3x³ - 8x² + 3x – 5 and q(x) = ax² - 3x + b is h(x) = 3x³ - 12x² + cx-13 then find the values of the constants a, b and c.

19. If the difference of p(x) = ax 45 + x 33 -  2x 25 + bx7 – 1 and q (x) = x 45 + x 34 - x 23 + x2 + 3 is h(x) = 15 x 4 + 112x3 115 x 2 + 115 – c then find the values of the constants a, b and c.
Solution:
Given p (x)= ax 45 + x 33 - 2x 25 + bx7 - 1
q (x) = x 45 + x 34 - x 23 + x2 + 3
According to question
     p (x) - q (x) = h(x)
⇒ ax 45 + x³3 2x²5 + bx7 – 1 - x 45 - x 34 + x 23 - x2 + 3 = 15 x4 + 112 x³ - 115 x² + 514 x - c
Comparing the coefficient
a5 - 15 = 15
⇒ a5 = 25 a = 2,
b7 - 12 = 514
⇒ b7 = 514 = -27
⇒ b = -2
And,
     -c = 3-1
⇒  c = -2.
∴ a = 2 , b = -2 & c = -2.