VIDEO: Calculate zeros by factoring out

Calculating zeros - what do you have to do?

• When it comes to the term "zeros", it is always a calculation that includes Functions has to do.
• The zeros of a function f (x) are exactly the places on the x-axis at which the function intersects them. There the function value, i.e. the y-value, is zero.
• The condition for a zero is always f (x) = 0.
• Depending on the function equation f (x), this condition results in different calculation steps with which you have to calculate the x values.
• In the simplest case you have to solve an equation for x (using known formulas and rules). For quadratic functions (Parabolas) you can use the pq formula, for example.


Factoring out - an explanation

Factoring out is a mathematical operation that can be used for many arithmetic tasks ...

Zeros in polynomials - this is how factoring works

Problems with calculating zeros often arise when the function is a polynomial, i.e. a completely rational function whose degree is greater than 2. Such a function is, for example, f (x) = x³ + 2x² - 1, which is of the third degree and cannot be cracked with the usual methods.

• One possible method of calculating zeros here, too, is to factoring out, which reduces the degree of the polynomial.
• However, these polynomials have to meet a very special condition: The term must not be a constant contain - or in other words: All components of the functional term must contain at least one "x" contain.
• The above example f (x) = x³ + 2x² - 1 cannot be solved by factoring out, but the function f (x) = x³ + 2x² can.
• In this case you proceed in such a way that you exclude as high a power of x as possible from the function term. This lowers the power of x in brackets, which is often easier to calculate.
• If you are to calculate the zeros for the function f (x) = x³ + 2x², then x³ + 2x² = 0, the condition, applies first.
• Now you factor out x² (the highest possible power) and get: x² (x + 2) = 0.
• This is a product. This product can only become zero if either the first factor (x²) becomes zero or the second factor (x + 2) becomes zero.
• In the first case you get x as the zero₁ = 0 (x² = 0 also follows x = 0).
• In the second case you get x as the zero₂ = -2 (calculated from x + 2 = 0).

Conclusion: In some cases the zeros of a completely rational function can be calculated by adding a Excluding the power of x and then separating the two parts of the function that have a lower degree treated.