Question

Factor the high-degree polynomial expression 7x2+42x+63=0

Answer 1

To factor the polynomial \(7x^2 + 42x + 63 = 0\), we can start by factoring out the greatest common factor (GCF) from all the terms.

1. Identify the GCF of the coefficients: The coefficients are \(7\), \(42\), and \(63\), and the GCF is \(7\).

2. Factor out \(7\):

\[
7(x^2 + 6x + 9) = 0
\]

3. Now, we can factor the quadratic \(x^2 + 6x + 9\). We need two numbers that multiply to \(9\) (the constant term) and add up to \(6\) (the linear coefficient).

The two numbers that satisfy this are \(3\) and \(3\):

\[
x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2
\]

4. Therefore, we can rewrite the polynomial:

\[
7(x + 3)^2 = 0
\]

5. To find the roots, we set \(7(x + 3)^2 = 0\):

\[
(x + 3)^2 = 0
\]

This gives us:

\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]

Thus, the completely factored form of the original expression is:

\[
7(x + 3)^2 = 0
\]

This indicates that the polynomial has a double root at \(x = -3\).