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All SAT Mathematics Resources

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Want to review SAT Mathematics but don’t feel like sitting for a whole test at the moment? Varsity Tutors has you covered with thousands of different SAT Mathematics flashcards! Our SAT Mathematics flashcards allow you to practice with as few or as many questions as you like. Get some studying in now with our numerous SAT Mathematics flashcards.

Flashcards: Working with Complex Polynomials

is a constant in the equation:

(x^2-kx+3)(x+2)=x^3-2x^2-5x+6

If the equation is true for all values of , what is the value of ?

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Flashcards: Working with Complex Polynomials

is a constant in the equation:

(x^2-kx+3)(x+2)=x^3-2x^2-5x+6

If the equation is true for all values of , what is the value of ?

Your answer:

Correct answer:

Explanation:

While this problem is certainly doable with pure algebra, many complicated polynomial questions like this one are much more quickly solved by picking numbers. There are lots of terms in this problem, but the problem specifically states that the equation must hold true for ALL values of so you can turn a lot of complicated algebra into some rather quick arithmetic by picking an easy value for . You can't pick x=0 because the only appearance of is when it's multiplied by , but you can pick x=1 leaving you with very small numbers to calculate. Doing so will leave you with:

(1^2-1k+3)(1+2)=1^3-2(1^2)-5(1)+6

And that simplifies quickly, to:

(4-k)(3)=1-2-5+6

Which becomes:

(4-k)(3)=0

And that means that must equal .

You could of course, do the algebra here, but as a general rule when the polynomials involve several multiplicative terms like this on the SAT, the question usually gives you an opportunity to pick a simple number to make the work much easier.

Flashcards: Working with Complex Polynomials

is a constant in the equation:

(x^2+2x+1)(x-3)=x^3-x^2+ax-3

If the equation holds true for all values of , what is the value of ?

-5

-7

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Flashcards: Working with Complex Polynomials

is a constant in the equation:

(x^2+2x+1)(x-3)=x^3-x^2+ax-3

If the equation holds true for all values of , what is the value of ?

Your answer:

-7

Your answer:

Correct answer:

-5

Explanation:

When polynomials have this many terms on the SAT, and when you see a phrase like "for all values of ," it is usually faster and easier to pick a number for to test which answer choices match the given information (that's typically when the question asks "which of the following expressions is equivalent to the above?") or, as in this case, to remove the commonly-occurring variable and solve for the less-common one. Here you have lots of terms but only one term, so since they say that this equation holds true for ALL VALUES of , you can pick an easy-to-use value like x=1 and then use arithmetic, not multi-step algebra, to solve for . If you plug in x=1 the given equation becomes:

(1^2+2(1)+1)(1-3)=1^3-1^2+1a-3

Which quickly simplifies to:

(1+2+1)(1-3)=1-1+a-3

And that simplifies to:

$4\times -2=a-3$

So -8=a-3 , meaning that a=-5 .

Note that you could of course do the algebra, distributing all the multiplication in the given equation and then combining like terms. But on most polynomial questions with this much multiplication to do, the shortcut of picking a small number can save you lots of time and be less error-prone.

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Flashcards: Working with Complex Polynomials

Flashcards: Working with Complex Polynomials

Flashcards: Working with Complex Polynomials

Flashcards: Working with Complex Polynomials

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All SAT Mathematics Resources

Flashcards: Working with Complex Polynomials

Flashcards: Working with Complex Polynomials

Flashcards: Working with Complex Polynomials

Flashcards: Working with Complex Polynomials

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