To factor the given polynomial [tex]\(21a^2 - 57a + 30\)[/tex] completely, let's follow a detailed, step-by-step solution.
### Step 1: Factor out the Greatest Common Divisor (GCD)
First, we notice that all the coefficients (21, 57, and 30) share a common factor, which is 3. So, we factor out 3 from the polynomial:
[tex]\[
21a^2 - 57a + 30 = 3(7a^2 - 19a + 10)
\][/tex]
### Step 2: Factor the Quadratic Expression Inside the Parenthesis
Next, we need to factor the quadratic expression [tex]\(7a^2 - 19a + 10\)[/tex]. To find the roots of the quadratic expression, we can use the quadratic formula:
[tex]\[
a = 7, \quad b = -19, \quad c = 10
\][/tex]
The quadratic formula is given by:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
### Step 3: Calculate the Discriminant
First, we calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac = (-19)^2 - 4(7)(10) = 361 - 280 = 81
\][/tex]
### Step 4: Calculate the Roots
Using the discriminant, we find the roots of the quadratic:
[tex]\[
x_1 = \frac{{-(-19) + \sqrt{81}}}{2 \cdot 7} = \frac{{19 + 9}}{14} = 2.0
\][/tex]
[tex]\[
x_2 = \frac{{-(-19) - \sqrt{81}}}{2 \cdot 7} = \frac{{19 - 9}}{14} \approx \frac{10}{14} \approx 0.7142857142857143
\][/tex]
### Step 5: Write the Quadratic in Factored Form
Now that we have the roots [tex]\(x_1 = 2.0\)[/tex] and [tex]\(x_2 \approx 0.7142857142857143\)[/tex], we write the quadratic in factored form:
[tex]\[
7a^2 - 19a + 10 = 7(a - 2.0)\left(a - 0.7142857142857143\right)
\][/tex]
### Step 6: Combine with the GCD Factor
Finally, include the factor of 3 we factored out initially:
[tex]\[
21a^2 - 57a + 30 = 3 \cdot 7(a - 2.0) \left(a - 0.7142857142857143\right)
\][/tex]
Simplifying, we get:
[tex]\[
21a^2 - 57a + 30 = 3(7a - 14)(a - 0.7143)
\][/tex]
We can distribute the constants:
[tex]\[
21a^2 - 57a + 30 = 21(a - 2)(a - 0.7143)
\][/tex]
Thus, the factored form of the polynomial [tex]\(21a^2 - 57a + 30\)[/tex] is:
[tex]\[
21(a - 2)(a - 0.7143)
\][/tex]
