Factor the high-degree polynomial expression 7x^2+ 42x + 63 = 0.
To factor the high-degree polynomial expression 7x^2 + 42x + 63 = 0, we begin by finding the greatest common factor (GCF) of the coefficients of the polynomial, which is 7.
Next, we divide each term of the polynomial by the GCF:
7x^2 + 42x + 63 = 7(x^2 + 6x + 9)
Now, we try to factor the quadratic expression inside the parentheses: x^2 + 6x + 9.
The quadratic expression can be factored as (x + 3)(x + 3), since 3 * 3 = 9 and 3 + 3 = 6.
Therefore, 7(x^2 + 6x + 9) = 7(x + 3)(x + 3).
So the factored form of the polynomial expression 7x^2 + 42x + 63 = 0 is 7(x + 3)(x + 3).
To factor the polynomial expression 7x^2 + 42x + 63 = 0, we can start by looking for common factors among the three terms. In this case, we notice that all three terms are divisible by 7. So, we can factor out 7:
7(x^2 + 6x + 9) = 0
Now let's focus on factoring the quadratic expression inside the parentheses. The quadratic expression (x^2 + 6x + 9) is a perfect square trinomial. It can be factored further as follows:
(x + 3)^2 = 0
Setting each factor equal to zero, we have:
x + 3 = 0
Solving for x, we find:
x = -3
Therefore, the factored form of the polynomial expression 7x^2 + 42x + 63 = 0 is:
7(x + 3)^2 = 0
And the solution to the equation is x = -3.
To factor the high-degree polynomial expression 7x^2 + 42x + 63 = 0, follow these steps:
Step 1: Check for a common factor.
In this case, since all coefficients are divisible by 7, we can factor out 7 from each term: 7(x^2 + 6x + 9) = 0.
Step 2: Factor the quadratic expression inside the parentheses.
To factor x^2 + 6x + 9, we need to find two numbers whose sum is 6 and whose product is 9. The numbers that fulfill these criteria are 3 and 3. Therefore, the quadratic expression factors as (x + 3)(x + 3), which can also be written as (x + 3)^2.
Step 3: Write the factored form of the equation.
Substituting the factored form of the quadratic expression into the equation, we have 7(x + 3)^2 = 0.
So, the factored form of the polynomial expression 7x^2 + 42x + 63 = 0 is 7(x + 3)^2 = 0.