Explain the steps in using the identity x^2 + (a + b)x + ab = (x + a) (x+b) to

Question

find the factored form of the polynomial x^2 + 10x + 21. Make sure you include the factored form of the polynomial in your answer. (3 points)

Answer 1

To factor the polynomial $x^2+10x+21$ using the identity $x^2+(a+b)x+ab=(x+a)(x+b)$, follow these steps:

1. Identify the coefficients: From the polynomial $x^2+10x+21$, we can identify:

   • The coefficient of $x$ (which is $a+b$) is $10$.
   • The constant term (which is $ab$) is $21$.

2. Set up the equations: We need to find two numbers, $a$ and $b$, such that:

   • $a+b=10$
   • $ab=21$

3. Find suitable $a$ and $b$: We can look for pairs of factors of $21$ that add up to $10$. The pairs of factors of $21$ are:

   • $(1,21)$
   • $(3,7)$

   Now we check which pair adds up to $10$:

   • $1+21=22$ (not suitable)
   • $3+7=10$ (suitable)

   Therefore, we have $a=3$ and $b=7$.

4. Use the identity: Using the values of $a$ and $b$, we can now express the polynomial in its factored form: $$x^2+10x+21=(x+a)(x+b)=(x+3)(x+7).$$

5. Final factored form: The factored form of the polynomial $x^2+10x+21$ is: $$(x+3)(x+7).$$

Thus, the factored form of the polynomial is $(x+3)(x+7)$.