SAT II Math II : Factoring and Finding Roots

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

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Example Question #541 : Sat Subject Test In Math Ii

A polynomial  of degree 4 has as its lead term  and has rational coefficients. One of its zeroes is ; this zero has multiplicity two. 

Which of the following is this polynomial?

Possible Answers:

Cannot be determined

Correct answer:

Explanation:

A fourth-degree, or quartic, polynomial has four zeroes, if a zero of multiplicity  is counted  times. Since its lead term is , we know by the Factor Theorem that

 

where the  terms are the four zeroes.

A polynomial with rational coefficients has its imaginary zeroes in conjugate pairs. Since  is such a polynomial, then, since  is a zero of multiplicity 2, so is its complex conjugate . We can set  and , and 

We can rewrite this as

or

Multiply these factors using the difference of squares pattern, then the square of a binomial pattern:

Therefore, 

Multiplying:

                            

                            

                       

       

Example Question #542 : Sat Subject Test In Math Ii

Find the roots to:  

Possible Answers:

Correct answer:

Explanation:

This equation cannot be factored.  Since this is a parabola, we can use the quadratic equation to find the roots.

Substitute the coefficients of .

The answers are:  

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