The Remainder Theorem is not something that you will use many times when taking the SAT, but it has shown up on a couple of problems in the practice tests that have been released in The Official SAT Study Guide.
The Remainder Theorem says that if you divide some polynomial p(x) by the linear factor x – a, the remainder that you get is equal to p(a). For example, consider the problem below where the polynomial p(x) = x² – 2x – 11 is divided by x – 5 using either Long Division or Synthetic Division.
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Earlier this week, we wrote a post on Completing the Square. Understanding this concept will help you with problems involving both circles and parabolas. If you’ve read through that post and you’re ready to test your skills, you’ll find five problems below that you can try. The solutions can also be found in the links below.
Complete the Square Problems
Complete the Square Solutions
If you have any questions about these problems, completing the square in general, or anything else to do with the SAT, send us an email at email@example.com .
When the new SAT came out in March 2016, it became clear that the test makers were looking to test some skills that hadn’t appeared on the previous version of the test. One of those skills is “completing the square,” which can be helpful when you are working on problems involving circles and parabolas.
So what is completing the square? Well, a perfect square trinomial x² + bx + c is one that can be rewritten in the form (x + k)², where k is some integer. For instance x² + 10x + 25 is a perfect square trinomial because is can be factored into (x + 5)(x + 5) = (x + 5)².
When you are asked to complete the square, you need to find that value of c (there is always only one possible value) that will make the trinomial a perfect square. For instance, suppose you were asked this question:
What number will complete the square for x² + 6x + ___ ?
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Now more than ever, with the focus of the New SAT Math test being on algebra and functions, you need to make sure that you are an ace at algebraic operations. This includes things like simplifying algebraic expressions, solving linear and quadratic equations, and solving for a variable in a formula.
At the bottom of the page, we have links to 10 SAT practice problems that involve these topics, along with their solutions.
But first, a few friendly reminders (things you’ll probably remember your algebra teacher saying many times over!):
Continue reading SAT Math: Algebraic Operations
The new SAT has shifted the focus of the math questions away from geometry and more toward algebra and functions. One of the topics you’ll want to master is systems of linear equations. If the practice tests are any indication, you can expect to see four or five of these questions on each test.
Below you’ll find a link to 10 sample system of equations questions (and a second link to the solutions).
But first, let’s do a little review.
Most systems of linear equations have a single ordered pair as their solution. It’s the point where the two lines intersect, and you can find the coordinates of that point by graphing the lines on your calculator or using the substitution or elimination methods. But what about the special cases? Let’s take a look at systems that have either an infinite number of solutions or no solution at all.
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