Question

Find the zeroes of the quadratic polynomial $x^2 + 7x + 10$, and verify the relationship between the zeroes and its coefficients.

Hint:

Always double check the signs when splitting terms and calculating zeroes.
A common mistake is forgetting that a factor of $(x+a)$ gives a zero of $x = -a$.

Answer 1

Step 1: Understanding the Question:
We are given a quadratic polynomial $p(x) = x^2 + 7x + 10$.
We need to perform two tasks:
1. Find the zeroes of this polynomial.
2. Verify the mathematical relationship between the zeroes and the coefficients of the polynomial.

Step 2: Key Formula or Approach:
1. To find the zeroes, we set $p(x) = 0$ and factorize the quadratic polynomial using the splitting the middle term method.
2. For a standard quadratic polynomial $ax^2 + bx + c$, let the zeroes be $\alpha$ and $\beta$. The relationship between the zeroes and coefficients is:
- Sum of zeroes: $\alpha + \beta = -\frac{b}{a}$
- Product of zeroes: $\alpha \cdot \beta = \frac{c}{a}$

Step 3: Detailed Explanation:

• 1. Identify coefficients from the given polynomial $x^2 + 7x + 10$:
Comparing with $ax^2 + bx + c$, we get:
\[ a = 1, \quad b = 7, \quad c = 10 \]

• 2. Find the zeroes by setting the polynomial to 0:
\[ x^2 + 7x + 10 = 0 \]
Split the middle term $7x$ into $5x + 2x$ (since $5 \times 2 = 10$ and $5 + 2 = 7$):
\[ x^2 + 5x + 2x + 10 = 0 \]
Factor by grouping:
\[ x(x + 5) + 2(x + 5) = 0 \]
\[ (x + 2)(x + 5) = 0 \]

• 3. Determine the zeroes:
- $x + 2 = 0 \implies x = -2$
- $x + 5 = 0 \implies x = -5$
So, the zeroes are $\alpha = -2$ and $\beta = -5$.

• 4. Verify the Sum of Zeroes:
- From the calculated zeroes:
\[ \alpha + \beta = -2 + (-5) = -7 \]
- From the coefficients formula:
\[ -\frac{b}{a} = -\frac{7}{1} = -7 \]
Since both values are equal, the sum of zeroes is verified.

• 5. Verify the Product of Zeroes:
- From the calculated zeroes:
\[ \alpha \cdot \beta = (-2) \cdot (-5) = 10 \]
- From the coefficients formula:
\[ \frac{c}{a} = \frac{10}{1} = 10 \]
Since both values are equal, the product of zeroes is verified.


Step 4: Final Answer:
The zeroes of the polynomial are $-2$ and $-5$. Both the sum of zeroes ($-7$) and product of zeroes ($10$) have been successfully verified against the coefficients of the polynomial.