While playing in a garden, Samaira saw a honeycomb and asked her mother about it. Her mother replied that it is a honeycomb. Also, she told her that the shape of the honeycomb formed is a mathematical structure. The mathematical representation of the honeycomb is shown in the graph.
Based on the above information, answer the following questions :
(i) How many zeroes are there for the polynomial represented by the graph given ?
(ii) Write the zeroes of the polynomial.
(iii) (a) If the zeroes of a polynomial x2+(a+1)x+b are 2 and 3, then determine the values of a and b.
OR
(iii) (b) If the square of difference of the zeroes of the polynomial x2+px+45 is 144, then find the value of p.
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Step-by-Step Solution
Step 1: Identify Zeroes from Graph
The zeroes of a polynomial are the x -values where the graph intersects the x -axis. By observing the given graph, we can see that the curve crosses the x -axis at three distinct points: x=−2, x=1, and x=3. Therefore, there are three zeroes for the polynomial.
Step 2: Solve Part (iii) (a): Find a and b using sum and product of zeroes
For a quadratic polynomial Ax2+Bx+C, the sum of the zeroes is given by -B/A and the product of the zeroes is given by C/A. Given the polynomial x2+(a+1)x+b and zeroes 2 and 3, we can set up two equations. The sum of zeroes is 2+3=5, which equals −(a+1). The product of zeroes is 2×3=6, which equals b. Solving these equations gives a=−6 and b=6.
Step 3: Solve Part (iii) (b): Find p using difference of zeroes
For the polynomial x2+px+45, let the zeroes be α and β. The sum of zeroes is α+β=−p and the product of zeroes is αβ=45. We are given that the square of the difference of the zeroes is 144, i.e., (α−β)2=144. We use the identity (α−β)2=(α+β)2−4αβ. Substituting the values, we get (−p)2−4(45)=144, which simplifies to p2−180=144. Solving for p, we find p2=324, so p=±18.