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

Step 1: Factor the quadratic polynomial

To find the zeroes of the quadratic polynomial, we need to set the polynomial equal to zero and solve for $x$. We can factor the quadratic expression by finding two numbers that multiply to $10$ and add up to $7$. These numbers are $2$ and $5$.

Step 2: Factor the expression

We rewrite the middle term $7x$ as $2x + 5x$. Then, we factor by grouping. We take out the common factor $x$ from the first two terms and $5$ from the last two terms. This results in the factored form $(x+2)(x+5)=0$.

Step 3: Find the zeroes

For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for $x$ to find the zeroes of the polynomial. The zeroes are $x = -2$ and $x = -5$.

Step 4: Identify coefficients

For a general quadratic polynomial $ax^2 + bx + c$, we identify the coefficients. In our case, for $x^2 + 7x + 10$, we have $a=1$, $b=7$, and $c=10$.

Step 5: Verify sum of zeroes

The sum of the zeroes of a quadratic polynomial $ax^2+bx+c$ is given by the formula $-\frac{b}{a}$. We calculate the sum of the zeroes we found, $-2 + (-5) = -7$, and compare it with $-\frac{b}{a}$. Both values are $-7$, so the relationship is verified.

Step 6: Verify product of zeroes

The product of the zeroes of a quadratic polynomial $ax^2+bx+c$ is given by the formula $\frac{c}{a}$. We calculate the product of the zeroes we found, $(-2) \times (-5) = 10$, and compare it with $\frac{c}{a}$. Both values are $10$, so the relationship is verified.

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