Question

While playing in a garden, Tanya saw a honeycomb and asked her mother what it was. Her mother replied that it is a honeycomb made by honey bees to store honey. 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.

1. How many zeros are there for the polynomial represented by the given graph?
2. Write the zeros of the polynomial. 
3. If the zeros of a polynomial x² + (p + 1)x + q are 2 and –3 then determine the values of p and g. 
4. If the square of the difference between the zeros of the polynomial x² + ax + 45 is 144 then find the value of a.

Answer 1

i. The upward-opening parabola cuts the x-axis at two distinct points, so it has two zeros.

ii. The graph cuts the x-axis at x = –7 and x = 7, so these are the zeros of the polynomial.

iii. For x² + (p + 1)x + q, the sum of zeros is –(p + 1) and the product of zeros is q.

Sum of zeros:

\[2+(-3)=-1=-(p+1)\]

\[p+1=1 \Rightarrow p=0\]

Product of zeros:

\[2\times(-3)=q \Rightarrow q=-6\]

iv. Let the zeros be α and β. 

Then α + β = –a and αβ = 45.

\[(\alpha-\beta)^{2}=(\alpha+\beta)^{2}-4\alpha\beta=144\]

\[(-a)^{2}-4(45)=144\]

\[a^{2}-180=144 \Rightarrow a^{2}=324\]

\[a=\pm 18\]