Digital SAT Math Practice Question #1193
Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.
Recall the Remainder Theorem. How can you use the given information about the remainder when P(x) is divided by (x-2) to find the value of 'k'?
Once you have the complete polynomial P(x), what theorem relates the factors of a polynomial to its roots (or zeros)? How can you use this to test each given option?
⚡ **Hack the Test: Graphing Calculator (Desmos/TI-84) & Backsolving** 1. **Find k first:** As in the standard solution, use the Remainder Theorem to find k = -7. So, P(x) = x^3 - 4x^2 - 7x + 10. 2. **Graphing Calculator:** Input P(x) = x^3 - 4x^2 - 7x + 10 into a graphing calculator (like Desmos or a TI-84). Look for the x-intercepts (where the graph crosses the x-axis). An x-intercept at x=a means (x-a) is a factor. * The graph will show x-intercepts at x = -2, x = 1, and x = 5. * Since x=1 is an x-intercept, (x-1) must be a factor. This immediately points to Option D. 3. **Backsolving (Smart Substitution):** Once you have P(x) = x^3 - 4x^2 - 7x + 10, you can quickly test the values that would make each factor zero. For example, for (x-1), test x=1. For (x+3), test x=-3. This is essentially what was done in the detailed solution, but framed as a 'hack' because you're using the options to guide your testing rather than deriving factors from scratch.
To solve this problem, we need to first determine the value of 'k' using the given remainder information, and then use the Factor Theorem to identify which of the options is a factor of the polynomial. **Step 1: Use the Remainder Theorem to...
Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...
Unlock Full Solution & 2,600+ Official Drills
Access detailed Socratic steps, trap breakdowns, authentic timed module simulations, and Desmos speed hacks for just $9.99/mo.
In a certain high school, the ratio of the number of juniors to the number of seniors is \( 3:2 \). The average (arithme...
Let $f(x) = \sqrt{2x + 6}$. What is the domain of $f(x)$?...
The absolute value inequality \( |kx - 12| \le c \) has a solution set of \( -2 \le x \le 8 \), where \( k \) and \( c \...
The function $p(t) = 150(1.5)^t$ represents population. By what factor does it grow when $t$ increases by 2?...