SAT Math : Polynomials

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #31 : Polynomials

If the polynomial 

is divided by 

,

what is the remainder?

Possible Answers:

Correct answer:

Explanation:

By the Remainder Theorem, if a polynomial  is divided by a binomial , the remainder is 

Let . Setting , if  is divided by , the remainder is , which can be evaluated by setting  in the definition of  and evaluating:

Example Question #5 : Factoring Polynomials

Which of the following is a factor of the polynomial  ?

Possible Answers:

Correct answer:

Explanation:

Call 

By the Rational Zeroes Theorem, since  has only integer coefficients, any rational solution of  must be a factor of 54 divided by a factor of 1 - positive or negative. 54 has as its factors 1, 2, 3, 6, 9, 18, 27 , 54; 1 has only itself as a factor. Therefore, the rational solutions of  must be chosen from this set:

.

By the Factor Theorem, a polynomial  is divisible by  if and only if  - that is, if  is a zero. By the preceding result, we can immediately eliminate  and  as factors, since 12 and 16 have been eliminated as possible zeroes.

 

Of the three remaining choices, we can demonstrate that  is the factor by evaluating :

, so  is a factor. 

Of the remaining two choices,  and , both can be proved to not be factors by showing that  and  are both nonzero:

 

, so  is not a factor.

 

, so  is not a factor. 

Example Question #7 : How To Factor A Polynomial

If the polynomial 

is divided by 

,

what is the remainder?

Possible Answers:

Correct answer:

Explanation:

By the Remainder Theorem, if a polynomial  is divided by a binomial , the remainder is 

Let . Setting  (since  ), if  is divided by , the remainder is , which can be evaluated by setting  in the definition of  and evaluating:

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