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In this lesson, we will learn how to find the number and type of roots of polynomials and how to find unknown coefficients if the roots are given.

Q1:

Determine the type of the roots of the equation ( 2 π₯ β 4 ) + 1 7 = 0 2 .

Q2:

If π + π π is a root of the polynomial π ( π₯ ) , what is the value of π ( π + π π ) ?

Q3:

Does the polynomial π π§ + π π§ + π π§ + π π§ + π π§ + π 5 4 3 2 , where π is nonzero and all the coefficients are real, have at least one real root?

Q4:

If π + π π is a root of the equation π ( π₯ ) = 0 , where π ( π₯ ) is a polynomial with real coefficients, which other complex number must also be a root?

Q5:

Is it possible for a polynomial with real coefficients to have exactly 3 non-real roots?

Q6:

How many roots does the polynomial ( 3 π₯ β 1 ) ( π₯ + 4 π₯ β 2 ) 2 3 have?

Q7:

How many real roots could the polynomial π ( π₯ ) = π π₯ + π π₯ + π π₯ + π π₯ + π π₯ + π 5 4 3 2 have given that π , π , π , π , π , and π are all real?

Q8:

Are the roots of the equation 3 π₯ + 2 4 π₯ + 4 8 = 0 2 real and different?

Q9:

Given that π ( π₯ ) = π π₯ + π π₯ + π 2 has a zero at 3 β 4 π and π ( 0 ) = 1 0 0 , determine the values of π , π , and π .

Q10:

If 7 and 6 are the roots of the equation π₯ + π π₯ + π = 0 2 , what are the values of π and π ?

Q11:

If the roots of the equation π₯ + 1 3 π₯ + π = 0 2 have a difference of 3, what is the value of π ?

Q12:

Find the quadratic equation whose roots are 9 + 7 π π and 9 + 7 π π 2 .

Q13:

How many real solutions does the equation 4 π₯ + 4 π₯ = β 1 2 have?

Q14:

Determine the type of the roots of the equation ( π₯ β π ) ( π₯ β π ) β 4 6 = 0 , if π and π are real numbers.

Q15:

Let π be a complex cube root of unity. Form a quadratic equation whose roots are οΊ 1 β ( 1 + π ) ο β 1 β 1 and ο» 1 β οΉ 1 + π ο ο 2 β 1 β 1 .

Q16:

Determine the type of the roots of the equation ( π₯ β 1 0 ) ( π₯ + 1 0 ) = 2 ( π₯ + 8 ) ( π₯ + 6 ) .

Q17:

Determine the type of the roots of the equation π₯ + 4 π₯ + 1 = 3 .

Q18:

Determine the quadratic equation whose roots are οΉ 2 + 2 π + π ο 2 3 and οΉ β 4 + 5 π β 4 π ο 2 3 .

Q19:

Find the quadratic equation whose two roots are β 4 1 + π and β 4 1 + π 2 .

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