find all the zeros of each function.
1. f(x)=2x^4-13x^3-34x^2+65x+120
2. f(x)=4x^4+19x^2-63
To find the zeros of each function, we need to determine the values of x that make the function equal to zero. This can be done by factoring the function or by using numerical methods such as the Rational Root Theorem or the graphing calculator.
1. f(x) = 2x^4 - 13x^3 - 34x^2 + 65x + 120:
Step 1: Factor out common terms, if possible.
Our function does not have any common terms that can be factored out.
Step 2: Use numerical methods or factor the equation.
Since factoring a quartic equation is generally difficult, we can use numerical methods or a graphing calculator to find the zeros. Alternatively, we can use the Rational Root Theorem to identify possible rational roots and then test them.
Using a graphing calculator, we can plot the function and find the x-intercepts, which correspond to the zeros.
Alternatively, we can divide the function by x - c, where c is a possible rational root, until we obtain a quadratic equation that can be factored or solved using the quadratic formula.
2. f(x) = 4x^4 + 19x^2 - 63:
Step 1: Factor out common terms, if possible.
Our function does not have any common terms that can be factored out.
Step 2: Use numerical methods or factor the equation.
Again, factoring a quartic equation is generally difficult. We can use numerical methods or a graphing calculator to find the zeros. Alternatively, we can use the Rational Root Theorem to identify possible rational roots and then test them.
Using a graphing calculator, we can plot the function and find the x-intercepts, which correspond to the zeros.
Alternatively, we can divide the function by x - c, where c is a possible rational root, until we obtain a quadratic equation that can be factored or solved using the quadratic formula.
In both cases, it is recommended to use numerical methods or a graphing calculator to find the zeros, as factoring quartic equations can be complex.