Final answer:
The question appears to involve factoring a quadratic polynomial "9x² + 11x + 2". One would find two numbers that multiply to 'ac' (18 in this case) and add to 'b' (11 in this case), then split the middle term and factor by grouping to obtain (x + 1)(9x + 2).
Step-by-step explanation:
Factoring Quadratic Polynomials
The question provided seems to be related to the factoring of quadratic polynomials. Typically, when factoring quadratics of the form ax² + bx + c, where 'a' is a constant greater than 1, one needs to find two numbers that multiply to 'ac' and add to 'b'. In our example, the polynomial to factor is 9x² + 11x + 2. To factor this, look for two numbers that multiply to 18 (9 x 2) and add to 11. The numbers 9 and 2 satisfy this criteria. Thus, we can split the middle term as follows:
9x² + 9x + 2x + 2
Grouping the terms gives us:
(9x² + 9x) + (2x + 2)
Factoring out the greatest common factor (GCF) from each group gives us:
9x(x + 1) + 2(x + 1)
Now, we can factor by grouping:
(x + 1)(9x + 2)
As a general rule, the power of a number can be added when multiplied with the same base, according to the formula x¹⁺¹ + x¹² = x¹(p+q).
Simplifying and Solving Quadratics
To simplify and solve quadratic equations like x² + bx + c = 0, you can use the quadratic formula if the polynomial doesn't factor nicely. The quadratic formula is −b ± √(b² - 4ac) / 2a. Making sure to simplify expressions and eliminate terms is crucial in simplifying algebraic equations.
For example, in the equation x² + 1.2 x 10⁻² x - 6.0 × 10⁻³ = 0, you would simplify the coefficients and then apply the quadratic formula if applicable.