Which of the following values of is a zero of
For SAT questions asking for a zero (root) of a polynomial and giving a small set of answer choices, it is usually fastest to substitute each choice into the function and see which one makes the expression equal 0. Work systematically down the list, compute carefully (especially with negative numbers and exponents), and stop as soon as you find a value that makes the function equal 0, rather than trying to fully factor the polynomial unless the factoring pattern is immediately obvious.
Hints
Recall the definition of a zero
A zero of a function is a number such that . Think about how you can check this using the answer choices.
Use substitution instead of factoring
Instead of trying to factor the cubic, substitute each answer choice into and see which one makes the result equal 0.
Be careful with signs and exponents
When you substitute negative values, carefully compute , , and step by step so you do not lose or flip any minus signs.
Desmos Guide
Define the function
In Desmos, type f(x)=x^3-4x^2-7x+10 to define the function.
Evaluate each answer choice
On separate lines, type f(-5), f(-4), f(-1), and f(1). Look at the outputs and identify which input makes the output exactly 0; that input is the zero of the function and matches the correct answer choice.
Step-by-step Explanation
Understand what a zero of a function is
A zero (or root) of a function is a value of that makes the function equal 0.
So we are looking for the value of among the choices that makes
equal 0, that is, we need .
Test choice A: x = -5
Substitute into .
Since , is not a zero of the function.
Test choice B: x = -4
Substitute into .
Since , is not a zero of the function.
Test choice C: x = -1
Substitute into .
Since , is not a zero of the function.
Test choice D and identify the zero
Now substitute into .
Here , so is a zero of the function. Therefore, the correct answer is choice D, .