Case Study Based Question on Polynomials
In a clubhouse of a residential society, four children Konark, Dhruv, Gary and Aman are playing a game using cards. They have prepared four cards and written the algebraic expression on their cards. They have shown the cards to each other. The card which Konark contain have the following expression 6x³+4x²+8x+4. When the expression is multiplied by 2 that information is written on Aman's card. Dhruv and Gary have written (4a-2b-3c)² and 6x²+17x+5 on their cards respectively.
Based on the above information answer the following questions:
(i) Write the expression for Aman's card and find its value at x=1/2. [2 Marks]
(ii) Factorise 6x²+17x+5 and write the zeroes of the polynomial. [2 Marks].
Doubt by Mehak
Solution :
Algebraic expression written on the card of Konark 6x³+4x²+8x+4
Aman 2(6x³+4x²+8x+4) = 12x³+8x²+16x+8
Dhruv (4a-2b-3c)²
Gary 6x²+17x+5
i) Expression for Aman's card is 12x³+8x²+16x+8
Let p(x) = 12x³+8x²+16x+8
Value of p(x) at x=1/2
p(1/2) = 12(1/2)³+8(1/2)²+16(1/2)+8
p(1/2) = 12(1/8)+8(1/4)+16(1/2)+8
p(1/2) = 12/8+8/4+16/2+8
p(1/2) = 3/2+2+8+8
p(1/2) = 3/2+18
p(1/2) = (3+36)/2
p(1/2) = 39/2
p(1/2) = 12(1/8)+8(1/4)+16(1/2)+8
p(1/2) = 12/8+8/4+16/2+8
p(1/2) = 3/2+2+8+8
p(1/2) = 3/2+18
p(1/2) = (3+36)/2
p(1/2) = 39/2
ii) Factors of 6x²+17x+5
By Spliting the middle terms
By Spliting the middle terms
6x²+17x+5
= 6x²+(15+2)x+5
= 6x²+15x+2x+5
= 3x(2x+5)+1(2x+5)
= (3x+1)(2x+5)
Zeroes of 6x²+17x+5
= 6x²+(15+2)x+5
= 6x²+15x+2x+5
= 3x(2x+5)+1(2x+5)
= (3x+1)(2x+5)
Zeroes of 6x²+17x+5
6x²+17x+5 = 0
(3x+1)(2x+5) = 0
3x+1 = 0
3x=-1
x=-1/3
(3x+1)(2x+5) = 0
3x+1 = 0
3x=-1
x=-1/3
2x+5=0
2x=-5
x=-5/2
2x=-5
x=-5/2
Other Important Question of Polynomial
[Class IX]